Finally got myself an SF and want to fly it this week.

Scenario:

Quickdraw is between two ships. Can she use the title to shoot both forward and backward when its her turn to fire and can she then do it again when she loses a shield?

As I read the Special Ops card it does not say once per round and though her pilot ability says once per round, that seems to only refer to triggering a primary attack so it seems legal that she can get 2 forward and 2 rear attacks in one round if the scenario presents itself.

Am I correct?

Yes, any time a TIE/SF with the Special Ops Training title gets to make an attack, it can trigger the extra attack through the title. IF it attacks from the primary (front) arc and chooses not to use the other title option (the extra attack die).