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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.

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.

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.