When your administrator has followed the Administrator's Guide to Display Profile Fields in Issues (v 2.0), you are able to search JIRA issues based on the user profile elements you configured.
We are using standard JIRA functionality for the search. Thus there are some basic principles you should be aware of when searching:
You cannot search profile field values if you haven't configured a mapping for that field. To set it up, follow the Administrator's Guide to Display Profile Fields in Issues (v 2.0).
You are searching the JIRA custom field your Administrator has configured. You are not searching the user profile fields directly. Therefore you have to use the name of the custom field for the search. The name might differ from the name of the user profile field.
The value of the custom field is always relative to a reference user field in the ticket. You might get results based on the user profile of the Assignee, the Reporter etc. This too depends on the configuration made by your Administrator.
If the search doesn't work as expected, please ask your Administrator for details on the configuration so you are aware of the current set-up and how it is supposed to work.