No.

The responsible party is the person placing content on the site.

Also, there is no trademark violation, so long as...

1) Person using the domain was using the term before someone else trade marked the name/phrase.

2) The domain relates to a different niche.

For example if the trade mark is "Best Stuff" for a soap.

Then the domain uses "Best Stuff" for Vegan recipes, there is no trademark infringement.

If you have questions, best to ask some expert about this.

Tip: Also, if you have Legal Shield coverage, very few people will ever have the budget to take you on in court.

Search Clarity for other questions involving Legal Shield for details about this service.