It always depends on the context that you are referring to. Are we talking about document library actions, or just arbitrary pieces of code calling the Repository-tier? Generally speaking, it is the responsibility of the client-side code (Share JavaScript) to show proper error messages. Most of the generic handling (i.e. document library actions) will simply show a generic error, unless you override the error handler via custom JavaScript.
Also, depending on the context of your Java code, some exception may simply not bubble up to the Share tier. This for example applies to transaction-level behaviours which will have any exceptions wrapped up in a very generic, technical exception, so the error message sent back to Share ultimately does not contain any useful information.
