I've received takedown claims and requests to settle before from some trolls (once by Getty but I didn't handle that one directly). Usually it was old contributed blog posts with an improperly cited image or site theme components we purchased that contained stuff that wasn't properly licensed. We'd take it down and then onto the next step...

I made friends with a few local lawyers, explained the situation and asked them to send a response on their letterhead or via email telling them in proper legal language to 'fuck off' (usually they'd mention it was outside statute of limitations and the use was never willful infringement or it was user contributed and we were only hosting the content). There was never a situation where they didn't leave us alone after that. I guess they assume real small companies will either panic and pay to settle or ignore legal summons and lose by default?