Do you call when you're the unknown traffic

Don't open the can of worms. Let them do what they are going to do without involving you. From experience, I have learned to play dumb.
What is the purpose of monitoring Approach (instead of talking to them)? They may point you out to other traffic, but you have to do the inverse calculation in your head to figure out where the other traffic is.
If you’re going to monitor frequency and do all of that, why not just get flight following?
In my case, same answer to both. Enhanced situational awareness while not really wanting to talk to anyone. Before someone accuses me of being afraid of ATC, when I'm not IFR, I've been a regular user of flight following since I've been a student pilot. My long solo cross country took me into an ARSA (now Class C). I'm a fan and a proponent of flight following. But the bottom line is, while I do have to aviate and navigate every time I fly, communicate is an option when VFR.

There are times I don't want to talk, just be aware. I don't see the "inverse calculation" problem @sarangan sees. Sure, if I were beamed into an airplane already in flight and suddenly heard a traffic call, I'd be disoriented, but that's not what's happening. Whenever I've done it and heard myself pointed out, It's been obvious (even before TIS-B).

And to answer the question in the thread, sometimes I call and identify myself. Sometimes I will also ask for FF. Sometimes I just keep quiet. It depends.
Funny thing I've noticed these days is that with ADS-B, I've had more than one controller call me in the blind when I am in "stealth mode" just to see if I was on frequency. I will always answer, but if I don't need air traffic service during a VFR flight, I probably won't make a call. But when I listen, if it appears that my speaking up and checking in would be helpful to the controller or the overall safety situation, I certainly will make my presence known. Otherwise I just enjoy the ride.......
Sometimes busy airports are too busy to pay attention to FF. During flight training my instructor and I flew from South Jersey to North Jersey (Greenwood Lake). We got flight following and handed off to New York. When we had Greenwood lake in sight we called to cancel flight following with no response. My instructor made the decision to land and call them on the phone. Somehow they forgot about us and appreciated the call.

Another time was flying south to Cape May and got FF from Atlantic City while flying along the coast. ATC diverted me east out over the ocean for incoming Coast guard helicopters. As the buildings got smaller as I headed out to sea I called ATC to request getting back yo my original route. They apologized for forgetting about me.

My take away is major airports are too busy with an onslaught of commercial traffic to really stay on top of little guys like me. I do use it for night flying and long cross country but always have a concern I will be lost in the ATC dealing with bigger fish.