LocalDateTime.atZone() means: Take this local time and consider it to be the local time in this time zone.

That means that 6:19 PM "local time" in Chicago is ... 6:19 PM in time zone EST, and that 6:19 PM "local time" in Tokyo is ... 6:19 PM in time zone JST.

When you call LocalDateTime.now(), you get the local time in your current default time zone, but the returned value don't know what time zone that is. As the javadoc says:

This class does not store or represent a time-zone. Instead, it is a description of the date, as used for birthdays, combined with the local time as seen on a wall clock. It cannot represent an instant on the time-line without additional information such as an offset or time-zone.

If you want to take the current real time and see what it is in Chicago and Tokyo, then you need to use either a universal time (Instant) or a time that know what time zone it represents (ZonedDateTime).

Using Instant

Instant now = Instant.now();
ZonedDateTime chicago = now.atZone(ZoneId.of("America/New_York"));
System.out.println("Chicago: " + chicago);
System.out.println("Chicago formatted: " + chicago.format(DateTimeFormatter.ofLocalizedTime(FormatStyle.FULL)));

ZonedDateTime tokyo = now.atZone(ZoneId.of("Asia/Tokyo"));
System.out.println("Tokyo: " + tokyo);
System.out.println("Tokyo formatted: " + tokyo.format(DateTimeFormatter.ofLocalizedTime(FormatStyle.FULL)));

Output

Chicago: 2017-11-05T20:02:45.444-05:00[America/New_York]
Chicago formatted: 8:02:45 PM EST
Tokyo: 2017-11-06T10:02:45.444+09:00[Asia/Tokyo]
Tokyo formatted: 10:02:45 AM JST

Using ZonedDateTime

ZonedDateTime now = ZonedDateTime.now();
ZonedDateTime chicago = now.withZoneSameInstant(ZoneId.of("America/New_York"));
System.out.println("Chicago: " + chicago);
System.out.println("Chicago formatted: " + chicago.format(DateTimeFormatter.ofLocalizedTime(FormatStyle.FULL)));

ZonedDateTime tokyo = now.withZoneSameInstant(ZoneId.of("Asia/Tokyo"));
System.out.println("Tokyo: " + tokyo);
System.out.println("Tokyo formatted: " + tokyo.format(DateTimeFormatter.ofLocalizedTime(FormatStyle.FULL)));

Output

Chicago: 2017-11-05T20:04:19.368-05:00[America/New_York]
Chicago formatted: 8:04:19 PM EST
Tokyo: 2017-11-06T10:04:19.368+09:00[Asia/Tokyo]
Tokyo formatted: 10:04:19 AM JST

Jesper has already given you short answers to your questions.

The no-arg method LocalDate.html#now obtains the current date from the system clock in the default (system) time-zone. Therefore, when you print LocalDate.now() in Madrid, India and New York, at the same time, you may get different results (depending on what time you do it).

Demo:

import java.time.LocalDate;
import java.time.ZoneId;

public class Main {
    public static void main(String[] args) {
        System.out.println(LocalDate.now(ZoneId.of("Europe/Madrid")));
        System.out.println(LocalDate.now(ZoneId.of("Asia/Kolkata")));
        System.out.println(LocalDate.now(ZoneId.of("America/New_York")));
    }
}

Output:

2022-09-28
2022-09-29
2022-09-28

Note: to understand this difference, check the output of the following program:

import java.time.LocalDateTime;
import java.time.ZoneId;

public class Main {
    public static void main(String[] args) {
        System.out.println(LocalDateTime.now(ZoneId.of("Europe/Madrid")));
        System.out.println(LocalDateTime.now(ZoneId.of("Asia/Kolkata")));
        System.out.println(LocalDateTime.now(ZoneId.of("America/New_York")));
    }
}

Result:

2022-09-28T21:03:56.438167
2022-09-29T00:33:56.443577
2022-09-28T15:03:56.444049

However, a LocalDate object does not store the time-zone information. Therefore, when you print an object of LocalDate (i.e. LocalDate#toString), the printed value will remain the same irrespective of the system time-zone.

A java.util.Date object too does not store the time-zone information. It stores the number of milliseconds from January 1, 1970, 00:00:00 GMT. However, when you print an object of java.util.Date (i.e. Date#toString), it uses the default (system) time-zone to print the value i.e. if you print an object of java.util.Date on systems with different time-zones set to them, you will get different results.

Since Java 8 the date/time API is pretty easy to work with.

In your case:

// your local date/time with no timezone information
LocalDateTime localNow = LocalDateTime.now();
// setting UTC as the timezone
ZonedDateTime zonedUTC = localNow.atZone(ZoneId.of("UTC"));
// converting to IST
ZonedDateTime zonedIST = zonedUTC.withZoneSameInstant(ZoneId.of("Asia/Kolkata"));

You'll see a difference in the time (and possibly the date) between zonedUTC and zonedIST, reflecting the time zone offset between the two.

Note the usage of withZoneSameInstant here, e.g. as opposed to withZoneSameLocal.

LocalDateTime is a date-time without a time zone. You specified the time zone offset format symbol in the format, however, LocalDateTime doesn't have such information. That's why the error occurred.

If you want time zone information, you should use ZonedDateTime.

DateTimeFormatter FORMATTER = DateTimeFormatter.ofPattern("yyyyMMdd HH:mm:ss.SSSSSS Z");
ZonedDateTime.now().format(FORMATTER);
=> "20140829 14:12:22.122000 +09"

Offsets vary over time

On April 6, 2020, most time zones on the east coast of North America such as America/Montreal and America/New_York use an offset of four hours behind UTC.

So, in those zones 11 PM on 2020-04-06 is simultaneously 3 AM on the 7th in UTC. Add four hours to 2020-04-06T23:00 to get 2020-04-07T03:00. Adding four hours brings us from the wall-clock time used in America/Port-au-Prince and America/Nassau to UTC.

The offset used by the time zones mentioned above are four hours behind UTC only half the year. The politicians in those locations have decided to observe Daylight Saving Time (DST) for about half the year. So half the year they jump their clocks ahead an hour for an offset-from-UTC of -04:00, and later in the year they fall back an hour for an offset-from-UTC of -05:00. For example, earlier in the year 11 PM in America/New_York would be 04:00 in UTC rather than the 03:00 time seen in April.

Standard time, in January, is five hours behind UTC. So, 11 PM plus five hours is 4 AM next day.

ZoneId z = ZoneId.of( "America/New_York" ) ;
ZonedDateTime zdt = ZonedDateTime.of( 2020 , 1 , 6 , 23 , 0 , 0 , 0 , zNewYork ) ; 
Instant instant = zdt.toInstant() ;  // 2020-01-07T04:00Z

Daylight Saving Time, in April, is four hours behind UTC. So, 11 PM plus four hours is 3 AM next day.

ZoneId z = ZoneId.of( "America/New_York" ) ;
ZonedDateTime zdt = ZonedDateTime.of( 2020 , 4 , 6 , 23 , 0 , 0 , 0 , zNewYork ) ; 
Instant instant = zdt.toInstant() ;  // 2020-04-07T03:00Z

By the way, DST is only one of many reasons politicians have for changing the offset used by the time zone(s) of their jurisdiction. Politicians around the world have shown a penchant for changing their offsets surprisingly often. Your programming should always expect a time zone to change its offset. Just because your particular zone of concern does not currently observe DST does not mean the offset will never change.

As for parsing 2020-04-06T23:00:00.000 as a LocalDateTime, you will get 2020-04-06T23:00:00.000. Ignoring time zones is the entire point of LocalDateTime. So no adjustments are made.

You may be thinking of LocalDateTime as representing a particular locality. But, no, it represents any locality or all localities. But never any one particular locality. For a particular locality, use ZonedDateTime.

Pseudo time zones

Another point: EST is not a time zone. Such 2-4 letter letter codes are not real time zones, are not standardized, and are not even unique. For example, by EST, did you mean Australian Eastern Standard Time or North American Eastern Standard Time? Is CST Central Standard Time or China Standard Time? Avoid these pseudo-zones in your date-time-handling.

And you happened to use the wrong pseudo-code. On April 6 2020, most of those east coast time zones are observing Daylight Saving Time (DST). So they would be considered to be in “EDT” rather than “EST”. Avoid the problem by specifying the real time zone name.

Real time zones are named using Continent/Region format. See Wikipedia for a list of real time zones.

Never call LocalDateTime.now

I cannot imagine a case where calling LocalDateTime.now would be the right thing to do. Determining the current moment requires a time zone (or offset). And LocalDateTime by definition has no time zone or offset. When a programmer writes LocalDateTime.now, you can bet they do not fully understand the necessary concepts.

When you call LocalDateTime.now, the JVM’s current default time zone is implicitly used to capture the current time as seen by the people in the region of that zone. And then the fact of that time zone is deleted.

This:

LocalDateTime.now() 

…is the same as this:

LocalDateDate.now( ZoneId.systemDefault() )

…which is the same as getting the current moment as seen in a particular time zone, followed by removing the time zone information:

ZonedDateTime.now( ZoneId.systemDefault() ).toLocalDateTime()

For more code examples demonstrating LocalDateTime.now, see the correct Answer by Ole V.V.

Where to use LocalDateTime

If LocalDateTime is not appropriate for getting the current moment, and is not appropriate for tracking any moment, what is the appropriate use for this class? Three things: representing any locality, representing all localities, and booking future appointments.

• Any locality would be something like stating when Christmas starts. This year Christmas starts at 2020-12-25T00:00 wherever you are in the world. Of course this means Christmas starts first in Kiribati after midnight, later in Japan after midnight, even later in Tunisia after midnight, and still later in Chicago after midnight.
• All localities would be something like stating our company policy that lunch breaks at all our factories in Delhi, Düsseldorf, and Detroit are scheduled for 12:30. So on April 6 2020, the break will be at 2020-04-06T12:30:00. This break will occur first in Delhi, several hours later in Düsseldorf, and even more hours later in Detroit.
• Booking future appointments where you intend to keep the same time-of-day regardless of changes to the time zone’s offset must recorded without the offset. If your next dental appointments is in six months at 3 PM, we want to record the 3 PM without regard for the offset. If the offset were to be changed by politicians, we still want the appointment to start when the clock strikes three on that date in that zone.

Set the appointment.

LocalDateTime appointment = LocalDateTime.of( 2021 , 1 , 23 , 15 , 0 , 0 , 0 ) ;

Determine a moment for producing a calendar. Every time you do this, you may get a different result if the politicians have changed the rules of this time zone.

ZoneId z = ZoneId.of ( "America/Panama" ) ;
ZonedDateTime dueToArrive = appointment.atZone( z ) ;

LocalDate

As for LocalDate, we saw in the examples above that the date depends on time zone. For any given moment, the date varies around the globe by time zone. It may be “tomorrow” in Tokyo Japan while still “yesterday” in Montréal Québec Canada.

So, you must specify a time zone when asking for the current date.

LocalDate ld = LocalDate.now( ZoneId.of( "Asia/Kolkata" ) ) ;

If you omit the time zone, the JVM’s current default time zone is implicitly applied. So LocalDate.now() becomes LocalDate.now( ZoneId.systemDefault() ). I recommend specifying your desired/expected zone explicitly, rather than rely on implicit default.

Server time zone

You said:

server which runs in EST vs UTC

FYI, servers should generally be set to UTC. Most of your thinking, programming, logging, and so on should be done in UTC. Learn to think of UTC as The One True Time, with all other zones but mere variations.

As a programmer, you should never rely on the default time zone. That is far outside your control. That default can be changed so easily by the user or sysadmin. Furthermore, any code in any thread of any app within the JVM can instantly change the JVM’s current default time with a call to TimeZone.setDefault. So even during execution of your app, the default can be changed at any moment.