One project manager has given three weeks to achieve a task which will potentially require all of the three weeks.
The second project manager has repeatedly assigned small tasks which have accumulated enough to set me back on the first deadline.
I might have the ability to tell one or the other "no I can't handle your assignments" right now but it would cause me a lot of anxiety in the future. I don't have the authority to tell a given PM no. Upper management seems to have no mechanism to assign my time to any particular local manager.
Yet I'm accountable for my time, and the PMs expect me to be able to answer for myself. But if I answer for myself, I poison the well with one PM and ostensibly our future relationship.
Any thoughts on how to solve this?
I've been in this situation before as a consultant, except in my case it was PMs repeatedly scheduling me for multiple on-site visits on conflicting dates. I didn't have the power to tell either one "no", but what I did have was the power to say "I have a conflicting assignment during that time, if you want me to do this one instead, you need to get [my manager] to reassign my priorities."
Unfortunately the fact that I could make them talk to somebody else never actually solved the underlying problem of each one insisting they had the right to schedule 100% of my time, but at least when those conflicts happened, I didn't have to resolve them myself.
If your direct manager isn't willing to referee this (up to and including telling those PMs to only assign you tasks through them if they can't get their act together between them), or you don't have a single direct manager to act as referee, or you're still being held responsible for the results of both projects by managers above yours, that's your cue that it's time to look for a new position, either internally or externally.