I don't know how concerned you are about the area surrounding these posts. We pulled ours out using a winch and a tractor bucket. It didn't tear anything up beyond about a 3' radius of it. If they are set in concrete, it will tear things up a little more and be harder to get out. The winch and tractor bucket worked for us, but the posts were not in our flower beds or lawn. Good luck!

sorry...I just noticed you mentioned the privacy fence being right next to it...is there a way to cut them off below ground level? Maybe with a torch? Make a mental note of where they are just in case you need to know later. Good luck!