In Germany most newer carts take 50ct, 1€, and 2€ coins. The older ones took 1€ (or 1DM before the introduction of the Euro). Abandoned carts on the premises aren't much of a thing anywhere I lived except for maybe the odd lazy person with a plastic coin in it (some.people carry those around e.g. on their key chain for when they don't have coins on hand).

The plastic coins are given out by the shops themselves.

I usually keep one in my pockets, not really out of laziness, I just don't use coins for much these days.

Any of the ones that take a 1 eur coin will take a 20c one as well. Which is good, because there are always a bunch of them in the cup holders.

I’ve never seen this anywhere, ever?

And honestly, if there was two euros going for simply moving a trolley back to its parking zone, I suspect you would have shoppers who found it worth doing that for all of the free carts, let alone people with less money coming in to collect.

Sometimes there are homeless people loitering in the parking lots offering to take your empty cart to pocket the coin in them.

Indeed. The only few cases are for trolleys left kilometers away.

Don't you have homeless people? ;) In Poland they are quite good at collecting coins from those (and also collecting cans).

What I more often see than carts with coins lying around is people straight-up stealing the carts. If they live in nearby streets they just push the cart home and then use it to carry or store stuff around the yard. You can see several such carts in the streets surrounding supermarkets.

And in Sweden we're removing the coins since the customers no longer carry cash...