I would try to figure out where your trespasser lives and pay them a visit and introduce yourself. I know, introverts nightmare! Maybe persuade your spouse to do it or go with you.
I think you’ll get a good read on the situation by meeting face to face. Start out with “Hi, we’re new to the neighborhood and wanted to introduce ourselves. We bought old Mr. John’s property and are going to build a house and barn; we’ll be on the property working on a regular basis.”
Go from there. If they express surprise and say they didn’t know the property had been sold, you can take a softer approach, like “I’m a little concerned about trespassers and stray dogs, but I don’t want to be the person who is new to the neighborhood and puts up a bunch of ‘No Trespassing signs.’ What do you think is the best way to let people know without ticking them off?” If they say, “Yes, we knew the property had sold but we’ve always walked our dogs there.” you might have to take a more aggressive approach.