Well, my tree is private and though I haven't DNA tested yet, I do plan to soon (I'm asking for a kit for Christmas). I have no plans to make my tree public when I get my results. The reason is that I know there are question marks and errors in my tree (which I am always working on) that I don't wish people to unknowingly duplicate so I only allow access after I've had a chance to explain any applicable grey areas to the person I'm giving access to. I have never denied anyone access when they've asked - I always encourage communication and collaboration with other researchers rather than grabbing things from trees without even talking to one another.
It is certainly not because I am a big meanie who wants to horde my information and not share it with anyone and I resent the implication and assumption that this is the attitude of anyone with a private tree. What is the problem with sending them a friendly message? There is no need to "grovel" - just say "Hey, I think we have a match in our trees and I was wondering if you're interested in collaborating?" Is communicating with other researchers really too much to ask?