They often (usually?) will leave some small change though.

Panhandlers are common targets of theft, which may explain stashing most of what they get in various places less exposed.

Yep. You don't want to be the guy with $100 in a cup held out in front of you. Remember, it's almost as easy to dip a bill out as it is to dip one in.

Yeah, there are a lot of factors into how much you leave in the pot. Grandparent's post was just one of the pressures to leave more money, though one that most people aren't aware of.

I like when they pick out the pennies from my donations and throw them on the ground. I then pick them up, because the pennies are valuable to me.

You have a home to stash the pennies in to save them up. Someone who lives on the street can't carry around pounds of pennies around until they have thousands of them to exchange for larger currency. Pennies might have value long-term, but they are worthless if you can't stash them in bulk.

Disclaimer: Canadian, we abolished pennies because they are worthless and a pain in the ass. ;)