Who is responsible for debt accrued while separated? How?

You would not be responsible for any of that she incurred during your time of separation.

Generally you would each pay your own.

Usually no. But this could depend on what the debt was accumulated for. It also depends on if your attorney has filed the appropriate documents and made the correct requests. A joint preliminary injunction enjoins the parties from accumulating further debt and purchasing expensive items. You must request the court issue an order for this.