If we were to ask the dogs, they would probably say that's what they do all the time!
So in a way it's up to us to learn what they are already saying. I've found that I can learn a lot just by observing dogs, not just our dog(s), when they play, walk, meet other people, etc. So their wishes are all there, just behind our eyes (or nose...).
If your dog wants to go out to the bathroom he will probably do something anyway, be more active, run to the door and back or whatever. So it would in principle be possible to add a cue that we find more clear (see below).
However we should remain a few things about dogs and how they learn:
dogs live in the moment without explicitly referring to actions in the past. Their level of "consciousness" is not the same as our;
as a consequence the learning is mostly simple associative learning. That means the way to teach them things is done by conditioning (classical or operant conditioning);
we should not adopt a point of view that is too anthropocentric and expect elaborate conscious processes.
The last point means: the dog won't elaborate a full scenario like we would when we want to obtain something from someone. To use the bathroom analogy again: the dog would first need to be fully aware of his need, then he would need to remember a similar situation in the past, and its full context (which is the difficult part), and he would then need to perform some specific action in order to make us let him go out.
Knowing the limitations can help us have realistic expectations and derive a proper way to obtain them.
In this case my answer would then be that yes, it is possible but primarily through conditioning. A few principles from the theory of learning, applied to dogs are useful in this case: reinforced behaviours increase in frequency, high probability behaviours tend to increase the frequency of low probability behaviours (Premack's principle).
Summarising my answer taking the example of a dog that wants to go out, but can't open/unlock the door (which is usually the case).
From the dog's point of view he's already telling you very clearly. But we need to add a cue which is more clear to us. For example ringing a bell or pulling on something, or simply barking in front of the door. The reinforcer here is you opening the door. That's the high probability behaviours, the dog want to go out and every time you open the door you reinforce him to do so. So you want to add a lower probability behaviour, ringing a bell next to the door. You can then train the dog to ring the bell, using the opening of the door as a reinforcer.
That way you used a mix of classical and operant conditioning: the stimulus is the feeling that the dog wants to pee and as he's house trained the response is that he wants to go out. The operant conditioning is that ringing the bell is a reinforced behaviour that is rewarded all the time. That's also a behaviour chain, were the first behaviour is ringing the bell, it is reinforced by another behaviour which is in turn reinforced by an actual reinforcer (primary reinforcer) in this case peeing in the garden.
As a side note this is a good opportunity to point out one of the advantage of positive reinforcement training (as opposed to punishment based training): we reward the dog for performing behaviours but we never inhibit behaviours with punishment. That means the dog always feels free to perform new behaviours (worst case they won't be reinforced and will extinct). The usual example is the 101 things to do with a box training session: you introduce a box to the dog and you reward him each time he does something. But the rule is: you reward only new behaviour. That's in a way teaching "creativity". The implication here is that if your dog is free to perform different kind of behaviour in front of you, most of the time you just have to choose what to reinforce. Imagine he wants his toy. If he start barking, you do nothing, then he'll try something else (say sit in front of you), you reinforce by giving the toy, and you achieve your goal.
This video ("Dog Training: How to Teach your Dog to ring a bell to be let outside") covers. I don't fully agree with the first part where he teaches the dog to ring the bell using a food reward. I think we can go directly with the actual reward, that will help the dog associate the bell with the door, and not ringing the bell as yet another trick.
Last thing: dogs can associate the present context with their action. For example they can probably associate that ringing the bell only works in your presence. However at first I would remove the bell when you're not there, to prevent the behaviour from being unrewarded.