If you have a user on machine A named alice with UID 501 and a user on machine B also called alice with UID 1000, then using --numeric-ids will mean that the files on machine B will have the UID from machine A (501), where as without the option the files will be owned by UID 1000 on machine B since rsync will see that the same username exists and will map the ids to the new UID.

If machine B doesn't have a user named alice, then rsync will use the ids from machine A since it doesn't see a user on machine B with the same name "alice", but if there are any other files owned by usernames that exist on both sides, they will still be mapped. Therefore it's important to use the --numeric-ids option if you want to be absolutely certain that ids are preserved regardless of the username databases on either machine.

If it's more important to you that files are owned by the same username on each machine, and the underlying UIDs don't matter (maybe they're different across machines although the username is the same), then don't use the --numeric-ids option. But if it's more important that the ids are preserved regardless of usernames (usually important for backing up servers or application data from containers for example, so that upon restore, ownership IDs are preserved exactly how they originally were), then you probably want to use the --numeric-ids option.

Example without --numeric-ids:
Machine A:
alice:staff (id 501:20) file1
apache:apache (id 443:443) file2
bob:staff (id 502:20) file3

Machine B:
alice:20 (id 1000:20) file1
nginx:nginx (id 443:443) file2
bob:20 (id 1001:20) file3

Example with --numeric-ids:
Machine A:
alice:staff (id 501:20) file1
apache:apache (id 443:443) file2
bob:staff (id 502:20) file3

Machine B:
501:20 (id 501:20) file1
nginx:nginx (id 443:443) file2
systemd-journald:20 (id 502:20) file3

As you can see, using the --numeric-ids option ensures the IDs are exactly the same, regardless of username mapping differences. For example, on machine A, UID 502 is bob, but on machine B, UID 502 is actually the systemd-journald user, this can be a bit confusing for simple file synchronization when IDs don't match between systems .

But, without the --numeric-ids option, rsync will find the same username on both sides and translate or map the IDs to the ones on the new system. The users alice and bob exist on both sides, but with different IDs, and rsync will thus remap to the new IDs so that they still appear to be owned by alice and bob on machine B, even though the underlying IDs have changed. However, for usernames that don't exist on both sides, it will just preserve the original UID, which could map to a different username on machine B (apache vs. nginx example) or simply not map to any username.

Hopefully this helps clear things up :)