How does the PO influence PB estimates?

When I was in the Product Owner role I would often work with the development team this was commonplace as we refined the backlog together. For example: 

If I had a user story that involved creating a queue task when we receive a document into our repository the development team may have an initial idea of what that estimate may look like. 

As we dive deeper into certain acceptance criteria their estimates would be impacted. I also wanted the tasks to be ordered by age based on the receive date, the age should be displayed, and anything sitting in the queue over 'x' amount of days (this had to be configurable by the business) would be highlighted to the users. 

This acceptance criteria created additional complexity and could be implemented incrementally. We ended up splitting the user story apart in order to implement the aging requirements at a later date. This insured I as the PO got what was most important at an earlier date....an automated queue task. 

It's really about communication with your development team and the willingness to let some things go for a period in time to get the most important functionality. 

Hope this helped! 

I think Tony Divel's response is good from the Product Owner's perspective. From a technical perspective, discussions between the Product Owner and Development Team can be about what technical debt to incur and how much to take on. A presented Product Backlog Item can be done with less effort or complexity, but perhaps making technical trade-off that would need to be paid back later. The Product Owner is in a position to determine if there is value in taking on some technical debt in exchange for faster feedback by getting something out a little faster or getting a couple more Product Backlog Items done.