If a creature has both persist and undying, which ability triggers when it dies with no counters?

When a creature with both persist and undying dies with no +1/+1 or -1/-1 counters on it, both triggered abilities trigger at the same time. This is because both conditions are met: the creature had no +1/+1 counters (satisfying undying) and no -1/-1 counters (satisfying persist). Per CR 603.2, all triggered abilities that trigger simultaneously are put on the stack at the same time.

The controller of the creature chooses the order in which these two triggers are placed on the stack (CR 405.3). The trigger placed on the stack last will resolve first. Whichever one resolves first will return the creature from the graveyard to the battlefield with the appropriate counter. When the second trigger then tries to resolve, the creature is no longer in the graveyard, so that trigger simply does nothing — it cannot find its target (CR 603.6d treats this as the ability failing to do anything because the object is in a different zone).

This means you effectively get to choose which counter the creature returns with. If you put the undying trigger on the stack last (so it resolves first), the creature comes back with a +1/+1 counter. If you put the persist trigger on the stack last (resolving first), the creature comes back with a -1/-1 counter.

Example: You control a 2/2 creature with both persist and undying and no counters. It dies. Both triggers go on the stack; you choose to order persist on top (resolving first). Persist resolves, returning the creature with a -1/-1 counter, making it a 1/1. Undying then resolves but finds the creature is no longer in the graveyard, so it does nothing.

In practice, returning with a +1/+1 counter (by letting undying resolve first) is usually better, so most players will stack the triggers to let undying resolve first. The key rules at play are CR 603.2 (simultaneous triggers), CR 405.3 (ordering triggers on the stack), and CR 603.6d (inability to find the object).