It is discussed here that it is generally okay to apply for multiple jobs in different ranks at the same university, but I want to know is it acceptable to apply two different postdoc positions in the same institution at the same time. I think I remembered seeing somewhere that you are not advised to do so because these two professors may talk to each other and find that you apply to both of them. Suppose I have sent an application to Prof. A but did not hear back, is it okay to send another application to Prof. B even if they are in the same institution or should I wait for a definite yes/no from Prof. A before moving on?
It is entirely reasonable to apply for multiple postions at any one time, even at the same institution, provided you do it appropriately. There are a number of considerations to digest, but pretty well most of what is discussed in your linked question is relevant.
The most important thing to consider is that the application you make is tailored for the position you are applying for. You need to be able to highlight what amazing skills and abilities you will be bringing to this partnership, and how it will benefit the project. You need to avoid a cold call approach, otherwise your applications will sound like spam. This means you need to change your cover letter a lot. It is likely, though, that the skills you bring to any project will be very similar. You are, after all, a product of your previous experiences. If you were to apply for two different positions, highlighting two very different sets of skills and abilities, I believe this could be to your detriment if this information was shared between prospective employers.
You need to weigh up the relative value of each position. If they are both about equal, then I would apply to both simultaneously. If, however, one was far more cherished than the other - one was your dream position with a triple Nobel Laureate with access to the best research facilities and guaranteed ongoing funding opportunities and endless publications and the other was with an also-ran professor (do they exist?) who publishes once in a blue moon, I'd probably hold back on the application for the latter, just in case.
However, if there are two positions being offered in the same institution that you are qualified and suitable for, I believe you'd be crazy to let the opportunity slip.
There is an added consideration in that if your two professors discuss your applications, they may realise the potential to be able to share your expertise. And let's face it - professors do discuss applicants.
No professor/academic/employer is going to be that precious to believe that they are the only person in the world that you are willing to work for. No, I take that back - I've met a couple who are like that, but you wouldn't want to work for them!