It turns really hot while the other has the same temperature as they both used to have. It also locks to the rotor in a two-step process: first the little play between the caliper and the mount disappears and then the pistons lock to the rotor. It doesn't happen to the other one where the pistons move more freely. So my conclusions are:
I need to clean the pistons and the caliper of the right side (the troubling side) caliper.
I need to examine if the heel of the brake lever (Chinese racing copy) is on all the time, if just by a little bit.
I need to change fluid (would need to do it anyways).

Edit:/
Of course this could be symptoms of the other caliper malfunctioning manifesting as a strain on the right side caliper...