Log in as another SQL Server user via SSMS to check their permissions?
You can run SSMS as a different user via runas /netonly as described here.

However for that method you would need to know the other user's password.

You can also use EXECUTE AS to impersonate that user from inside a query window logged in as you (as long as you have the required permissions to do so).

EXECUTE AS 'user_name';

/* Some SQL Statements */

REVERT;


Or you could adopt a policy of never granting permissions to users directly but always granting them to Windows groups or database roles. Then you just need to have another dummy user that you can grant the same role membership to.