Do Bayesian methods imply a choice of loss function? I think it's a framework for observational statistics, not for policy making or decision theory. You can use it as a tool during policy or decision making, where you should indeed have a debate about picking a loss function that corresponds to your goals.
If you're reporting point or interval estimates (rather than the entire posterior) then you are implicitly or explicitly optimising some kind of loss function.
Also, worth a reminder that continuous estimation problems can be viewed as decision problems too, it's not just about discrete decision-making.
I don't think it's great to take the view that, because I'm not making a decision based on this estimate myself, I don't need to worry about which loss function I'm implicitly optimising for when choosing an estimator. Someone else may need to.
