Problem: A user with full permissions within a database cannot see any profiling events when the user has full permissions to two or more AS databases and is not a server admin. The user will be able to trace any query related events for the first database via Profiler but he/she cannot capture any similar events for the other databases.
Reason: This is a known limitation because the trace is normally a feature restricted to server Admins. DB admins can only see the trace output whenever it is safe to assume from a security perspective that the events produced are for a specific database only. If its not possible to guaratee events related to a single databse, then the events are not exposed to the user.
Alternative Solution: An alternative solution to this problem is to go ahead to create multiple profiler traces, one for each database. When you connect to SSAS from the Profiler, you will be able to choose to profile any of the databases that you have full admin access. In this case, you will open up multiple profilers, one for each database and now you are profiling the AS databases desired.
Thanks to contributions by John Lam, Edward Melomed!