ArrayList already have support for this, with the equals method. Quoting the docs

... In other words, two lists are defined to be equal if they contain the same elements in the same order.

It does require you to properly implement equals in your MyData class.

Edit

You have updated the question stating that the lists could have different orders. In that case, sort your list first, and then apply equals.

You could use containsAny method from CollectionUtils (Apache Commons):

if(CollectionUtils.containsAny(listA, listB)){
    break;
}

There are certain reasons I would not prefer your approach. These are:

1) There are multiple method calls which reduces the readability.

2) You are using filter twice which decreases the performances. You could do it inside the same filter like I've shown below

3) With this approach you don't have a consolidated List which contains common elements. For that you need another else condition which increases the cyclomatic complexity.

4) A null check which could produce NPE if not handled carefully as you are using findFirst() which returns Optional (Although null can be replaced with default new Person("","") object which is again not recommended).

5) Finally don't use ArrayList<Person> originalPeople = new ArrayList<>();; rather declare as List, to follow programming to interface.

Rather I would use below approach:

private static List<Person> getPersonInList(
            final List<Person> newPeople, List<Person> originalPeople) {
        List<Person> list = new ArrayList<>();

        newPeople.forEach(p ->
                originalPeople.stream()
                              .filter(p1 -> p.getFirstName().equals(p1.getFirstName()) && 
                                      p.getLastName().equals(p1.getLastName()))
                              .forEach(list::add));
        return list;
    }

In short: there is exactly one assert that one needs when writing JUnit tests: assertThat

You just write code like

assertThat("optional message printed on fails", actualArrayListWhatever, is(expectedArrayListWhatever))

where is() is a hamcrest matcher.

This approach does work for all kinds of collections and stuff; and it does the natural "compare element by element" thing (and there are plenty of other hamcrest matcher that one can use for more specific testing; and it is also pretty easy to write your own matchers).

( seriously; I have not seen a single case where assertThat() would not work; or would result in less readable code )

You could overwrite Equals and HashCode of CustomObj. Then use Contains() to test uniqueness.

Point 1: See Kayaman's comment (wrapper class providing equals function).

Point 2: This wrapper class might additionally provide a lessThan function imposing an absolute order on the elements. You then could sort both lists and afterwards compare them element by element. This is more efficient than trying to identify each element of one list in the other one and additionally handles problems with duplicates elegantly.

Implementation examples for e. g. quicksort can be found all over the internet, e. g. here.

If you want to get the same value from two list, I give you a string example:

/**
 * @param args
 * @throws IOException
 */
public static void main(String[] args) throws IOException {
    // create a script engine manager

    List<String> list_1 = new ArrayList<String>();
    list_1.add("a");
    list_1.add("b");
    List<String> list_2 = new ArrayList<String>();
    list_2.add("a");
    list_2.add("b");
    list_2.add("c");
    System.out.println(getSameVaue(list_2,list_1));
}

public static List<String> getSameVaue(List<String> list1,
        List<String> list2) {
    List<String> result = new ArrayList<String>();
    if (list1.size() > list2.size()) {
        for (String s : list1) {
            if (list2.contains(s)) {
                result.add(s);
            }
        }
    } else {
        for (String s : list2) {
            if (list1.contains(s)) {
                result.add(s);
            }
        }
    }
    return result;
}

But if you want to use your Object, please do not forget implement equals method.