According to the TOS with the registrar, it is the registrants obligation to make sure there are no trademark violations with the domain they register. So yes, the registrant must make an effort to check for trademarks.
I believe if you read the registration agreements carefully, you will find that the registrant is required to assume liability for any TM problems, and that the domain registrant represents that he/she is not aware of any. There is no positive duty to go looking. But there is a fine line between that and remaining willfully ignorant. I can walk around town blindfolded and shooting a gun, but it makes it hard to argue that I wasn't trying to shoot anyone.
In the normal application of US trademark law in a judicial proceeding, and indeed as expressly stated under US trademark law, registration provides constructive notice of marks. "Constructive notice" is legalese for "even if you didn't know, you should have known, and we are going to treat you as if you knew".
The UDRP was intended to apply to intentionally abusive domain name registrations. Now, when we are talking about specific intent, we do not normally apply a standard of constructive notice. Intentional behavior, otherwise known as willful behavior, requires actual knowledge.
Panels often split on the issue of constructive notice of marks. For example, if both entities are located in the US, then a panel is more likely to apply a constructive notice standard than if the parties are located in different countries.
WIPO summarizes constructive notice issues this way:
3.4 Can constructive notice form a basis for finding registration and/or use in bad faith?
Most panels have declined to introduce the concept of constructive notice into the UDRP. However, where a complainant had a United States registered trademark and respondent was located in the United States, this concept has been used in a few cases to support a finding of registration and/or use in bad faith. In those cases, where the complainantâs trademark registration preceded the respondentâs domain name registration, respondent was presumed to have notice of the trademark.
Relevant decisions:
Toronto Star Newspaper Ltd. v. Elad Cohen DTV2000-0006 , Denied
Kate Spade, LLC v. Darmstadter Designs D2001-1384 , Transfer
Alberto-Culver Company v. Pritpal Singh Channa D2002-0757 , , Denied
Sterling Inc. v. Sterling Jewelers, Inc and Domain Traffic D2002-0772 , Transfer
The Sportsmanâs Guide, Inc. v. Modern Limited, Cayman Islands D2003-0305 , Transfer