I believe you can create a user account in that database, using WITHOUT LOGIN. This will create it where you can grant execute as permissions to the other users for any particular stored procedure in that common database. They don't have to use a password for it so you control it by who can EXECUTE AS with it. Here is a pretty good write-up on how it could be used, near the end he talks on how it can be used outside the context of the database.
From the comments I see you removed the default time limit like I mentioned but still the analysis didn't happen in enough time or ever complete as I feared.
18GB is definitely the largest trace I have ever heard of attempting to be analyzed by the Database Engine Tuning Advisor. I suggest you break your trc file up into smaller sizes. Either by looking at the data a bit more filtered or for a smaller window of time while still maintaining enough of a workload to capture both data modification and selects.
Alternatively -
I haven't been a big fan of the DETA and sometimes the results, even with a properly sized workload file, fly in the face of reality (most index suggestions are very wide covering index in a lot of cases, redundant indexes in overlapping column coverage at times, etc.) I would suggest a "Top 10" style tuning exercise may be a better approach. Find the worst performing queries by duration and/or reads and work on tuning them through query tuning and index tuning/design best practices.
Edited: Added a couple more tips based on comment from OP
I would also look to set an end time. DETA isn't supposed to be a "done in 5 minutes" deal. It is trying to be incredibly thorough and look at your database as deeply as it can and analyze the workload as long as it can to come up with the recommendations. Have you tried setting an end time for the analysis?
I'd check out this link on MSDN for some other tips - https://blogs.technet.com/themes/blogs/generic/post.aspx?WeblogApp=sql_server_isv&y=2011&m=04&d=08&WeblogPostName=fundamentals-running-database-engine-tuning-advisor-and-selecting-indexes&GroupKeys=
Bottom line though - This will be a slower operation. The larger the workload file and more complex the database, the slower it will be. If you are running on the same system that other work is going on then you are each competing for resources.
Best Answer
i doubt that
the ALTER TRACE permission is a server level permission, and access is at the server level; if a user can start a trace, he or she can retrieve event data no matter what database the event was generated in.
for more info
http://technet.microsoft.com/en-us/library/cc293611.aspx
http://msdn.microsoft.com/en-us/library/ms187611.aspx