Does this concern pages that were cached by the content view caching system? See http://ez.no/developer/articles/ez_publish_performance_optimization_part_3_of_3_practical_cache_and_template_solutions/caching_overview. If you want to put user specific information into node views, then you need to disable content view caching.
ezwebin on eZ Publish 3.9.0 had a issue where it caches pagelayout header (using cache block) pr users with same rights, and not pr user. So your not logged in as another user, it's just another users user name that shows up on the webpage( so no security issue besides seeing the name of another user an his user id in the markup).
You can update ezwebin to 1.2 to get the fix, but you'll need to update eZ Publish as the updated uses nested cach-block's witch didn't work on 3.9.0 (fixed in a later 3.9.x version so use latest 3.9.x version or newer).
Thank you very much for your help! Disabling templatecache in site.ini.append solved my problems
Regarding the issue in 3.9.0, it sounds exactly like the problem, but when I used {def $current_user=fetch( 'user', 'current_user' ) } in a template file it also gave the wrong user. And if the user was trying to change personal settings while the wrong name was shown it stated that the apporiate rights was missing for that account.
fetch current user issue, see Kristof's post. No rights to edit: This is caused by the fact the user id is in the url, so if you get wrong name you also get wrong url.