Grant access for any logged in user for common database
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.