Here’s where the suggestions are collected.
No, it’s completely server-side and has to stay that way to prevent massive code duplication.
The server does all the checking. The client just displays the results, in some cases filtering spelling errors which the user has added to their personal dictionary.
Maybe there’s some confusion here: if we have small data, users can download it and use LT offline. That’s a good thing. But when you mention client/server above, “client” to me means “add-on in some software that uses the HTTP API”, not “stand-alone version of LT”.