You choose the two target permanents as Nesting Grounds's second ability is put onto the stack. You choose which kind of counter to move as that ability resolves.

If either permanent becomes an illegal target, no counter is removed or put.

The two target permanents don't have to share a type, which can result in some counters on permanents that would not occur normally, such as loyalty counters on creatures or +1/+1 counters on lands. Keyword counters will grant the permanent that keyword even if it's meaningless (such as trample on an enchantment); +1/+1 counters won't affect the permanent unless it's a creature; and many named counters (such as soul counters) won't have an effect unless the recipient permanent refers to them in some way.

To move a counter from one creature to another, the counter is removed from the first permanent and put on the second. Any abilities that care about a counter being removed from or put onto a permanent will apply.