I've never tested/used BuddyPress with Event Organiser, however someone has made this plug-in: https://github.com/christianwach/bp-event-organiser which integrates the two. (And there are no known conflicts).
The only issue with the above mentioned plug-in, and this will relate to your other comment, is that the calendar is cached (but no per-user). This makes attempts to have per-user calendars buggy. This is something I'll address in the next update (adding a filter to change the cache key).
There's no in-built feature for displaying a per-user calendar, but I suspect that the above plug-in might enable you to do that. With Pro there is a 'user is attending' option for the calendar to display a calendar of events that user has purchased tickets for.