Brickipedia's CheckUser policy outlines the circumstances in which the CheckUser extension can be used. CheckUsers can be contacted by Special:Emailuser for private requests.
CheckUser may only be used to determine a user's IP address if there is significant behavioral evidence to warrant a check, or if a user who is suspected of using sockpuppets is being disruptive. Users will not be checked for reasons such as similar usernames, involvement in similar areas of the encyclopedia, etc. This principle is enforced by the other CheckUsers, and at any given time a CheckUser can come under community review for suspected improper use of the extension.
For a guide on how to use the CheckUser extension, see here.
Any user is entitled to know whether or not their account has been checked.
IPs are never publicly revealed, and the CheckUser log is kept private, so that only CheckUsers can view it. If there is ever a concern over privacy in regards to a CheckUser, it is possible to start a community review of them on the administration forum.
Any concerns over this policy can be discussed here.