Hi Jenni,
sorry for the slow reply, I am holiday for this moment. Let me try to help you.
The answer is a boring "it depends". How are the teams structured? If the teams are components ones, it will be tough to split the work, and I would suggest re-structure the teams as soon as possible. If the teams are feature teams or product means, in other words, they can have responsibilities from end-to-end. Then, it will be easier.
The point is the following, the Product Backlog should contain the items that will enable the product to get closer to the Product Vision. However, this can be quite abstract. What I've seen working is having clear approaches to pursue and then share with each team an end-to-end responsibility. The Backlog is unique, but it's key to ensure each team can still have the possibility to measure the impact of their work, and then, inspect and adapt.
To summarize, it's not about who does what, but which goals each team will be accountable for.
I hope that helps.