Digital delays allow you to do all sorts of evil things to your sound, and they're set up in all sorts of different ways. Most modern digital delays have two inputs and at least two outputs, so you can do:
Mono input, split to two signals each with its own delay engine and blend of dry/wet, possibly mixed back together before output.
Dual input, mixed, delayed and output
Dual input, kept separate with separate processing
Technically a ping-pong delay would be a special case of type one, with a single delay engine, and an LFO driving the panning of the output signal.
If you really want to blow your mind, a couple of interesting devices include the Akai Headrush which has 5 outputs that it sends the signal to, sequentially for a ping-ping-ping-ping-pong kind of thing. And the Yamaha UD Stomp that includes 8 separate delay engines that can be set up in parallel or daisy chained in any combination then output to stereo for very complex ping-pong effects.