Should every user story have an acceptance criteria?

Why does it matter who writes any of it?  If the problem statement is captured in a way that everyone understands it and the expectations are clear as to how to determine a solution is delivered, does it matter who wrote any of the parts?  

The act of refinement will usually lead to changes.  So in the end, everyone involved in the refinement process will be included in those that write the story.  Who originates the story is the individual that has identified a problem that needs to be solved.  

I will add that user stories are not part of Scrum.  They are never mentioned in the Scrum Guide.  Product Backlog items can capture the necessary information in any format that the Scrum Team decides is adequate to properly state the need. I have worked with teams where every Product Backlog Item was written differently because of the need to communicate different things. Don't feel that you have to use any specific format.  Use what is best to capture the what the Scrum Guide describes is to be contained in the Product Backlog.

The Product Backlog is an emergent, ordered list of what is needed to improve the product.

thank you so much. I thought user stories were part of scrum. helped me a lot

I'm reading several agility materials and I think I'm a little confused. After all, are user stories and acceptance criteria written by the PO or the development team?

A user story is a placeholder for a conversation about a potential requirement. The important thing is the conversation it enables: a good user story is not necessarily written at all.