As a project coordinator, would it be acceptable if I asked the employee "Would this be a reasonable deadline for you?"
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In short, yes and no. Yes, because the team member will be responsible for the deliverable. No, because a team member can only provide you estimates. In your question, I'm assuming you used the term deadline as estimate, don't you?
Thinking of estimates, you're not only allowed to, but you must ask your team for estimates.
An estimation (as the name says) is when something is expected to be available. A deadline, on the other hand, is the due date that the team must commit to.
Estimates comes from your team to you, deadlines comes from the stakeholders to you. It's your duty to make things match as much as possible, tailoring business need / work effort throughput to reach the perfect balance.
Two different processes done by different roles. Organizations, in my observations, typically combine them, which is mistake. Estimating should be done, as Tiago indicates, by the workers who now the work intimately, while the targeting should be done by the business, e.g., the people accountable for success. Estimating should be probabilistic, e.g., I can finish this between 3 and 7 days, most likely 5. Targeting should be deterministic, e.g., the deadline is 4 days.
The factors and variables that are under consideration in both processes are different and many are in direct conflict. This is why it needs to be two separate processes done by the two different roles; else, you will lose valuable considerations. For example, the workers will care less about winning the work with an aggressive price set using aggressive targets. The workers will care more about coming in on time and will target for themselves more safely. (This, of course, is not always true in all cases; it is just an example of a common scenario.)
If the deadlines are no the critical path (which I assume most of them would be) then I would ask people to commit to those dates. Anything on the critical path is important enough to get commitment for. There may be exceptions where the task and deadlines are based on estimates in which case people might not want to connect fully, but at least you want them to understand the urgency of delivering on those deadlines.