A great way to learn about DNA is through the exhibits Kathleen Zellner published with her motions for the Steven Avery case. I'll get a link for those, or Missy may have one handy.
We don't always leave DNA. And no one, to my knowledge can say conclusively, why we sometimes leave it and sometimes we don't. Hygiene, it seems, can be a factor. So if you wash your hands a lot, you will likely leave less DNA than someone who rarely washes their hands.
What I suspect in the encounter between Joey and Chase on the 4th is, that before eating lunch that day both men washed their hands. So DNA from others they touched would either have been diminished or washed off.
They had a lunch that lasted a few hours, it would seem. And in that time they may have touched their face, or wiped their hands on pants, built up perspiration, who knows. At the end of the lunch they shake hands, or bro-hug, or pass off an object to each other. Say, Joey hands Chase checks, Chase hands Joey a cup, or paperwork-this may never be known for certain.
Joey gets in his Trooper, now Chase's DNA is on his hands, and there is just enough for DNA to be left primarily on the steering wheel, in slight amounts on other objects. And again, because we don't always leave DNA, not everything Joey touches gets DNA on it (one of the reports I posted mentions that the more pressure applied to an object, the more DNA left). This theory would explain why there is substantially more DNA left on the steering wheel, because when driving that is where the driver would place the most pressure.
There was the question as to why when Chase was in the Trooper 6 weeks prior, his DNA was not left on the passenger side of the Trooper. That could be because he was in one of those paint-ball outfits, and kept his gloves on. Or again, after playing paint ball he relieved himself, and washed his hands just prior to getting in the Trooper.
Lots of variables with this stuff.