The puzzle about belief is that if I differ from you fundamentally in belief it makes us in a sense different persons. The grounds of our beliefs will be different and will have a circular relation with the beliefs themselves. Because I cannot challenge your grounds from my grounds, without circularity, you and I have literally different worlds. Wittgenstein says something like this in the Tractatus, though I do not think it does not alter 'the facts' as well - indeed it is the entanglement of 'the facts' with our life-and-being stance that is the mystery:
6.43 If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts - not what can be expressed by means of language. The world of the happy man is a different one from that of the unhappy man.
So I can only communicate with you in superficials; in fundamentals we literally do not communicate. In practice we struggle with this at the limits of our awareness and find ways to partly connect somehow, which also accounts in part for how desparately we cling to the foundations of commonsense and the public world, of which Hume writes so poignantly:
Having thus given an account of all the systems both popular and philosophical, with regard to external existences, I cannot forbear giving vent to a certain sentiment, which arises upon reviewing those systems. I begun this subject with premising, that we ought to have an implicit faith in our senses, and that this wou'd be the conclusion, I shou'd draw from the whole of my reasoning. But to be ingenuous, I feel myself at present of a quite contrary sentiment, and am more inclin'd to repose no faith at all in my senses, or rather imagination, than to place in it such an implicit confidence. I cannot conceive bow such trivial qualities of the fancy, conducted by such false suppositions, can ever lead to any solid and rational system. They are the coherence and constancy of our perceptions, which produce the opinion of their continu'd existence; tho' these qualities of perceptions have no perceivable connexion with such an existence. The constancy of our ]perceptions has the most considerable effect, and yet is attended with the greatest difficulties. 'Tis a gross illusion to suppose, that our resembling perceptions are numerically the same; and 'tis this illusion, which leads us into the opinion, that these perceptions are uninterrupted, and are still existent, even when they are not present to the senses. This is the case with our popular system. And as to our philosophical one, 'tis liable to the same difficulties; and is over-and-above loaded with this absurdity, that it at once denies and establishes the vulgar supposition. Philosophers deny our resembling perceptions to be identically the same, and uninterrupted; and yet have so great a propensity to believe them such, that they arbitrarily invent a new set of perceptions, to which they attribute these qualities. I say, a new set of perceptions: For we may well suppose in general, but 'tis impossible for us distinctly to conceive, objects to be in their nature any thing but exactly the same with perceptions. What then can we look for from this confusion of groundless and extraordinary opinions but error and falshood? And how can we justify to ourselves any belief we repose in them?
This sceptical doubt, both with respect to reason and the senses, is a malady, which can never be radically cur'd, but must return upon us every moment, however we may chace it away, and sometimes may seem entirely free from it. 'Tis impossible upon any system to defend either our understanding or senses; and we but expose them farther when we endeavour to justify them in that manner. As the sceptical doubt arises naturally from a profound and intense reflection on those subjects, it always encreases, the farther we carry our reflections, whether in opposition or conformity to it. Carelessness and in-attention alone can afford us any remedy. For this reason I rely entirely upon them; and take it for granted, whatever may be the reader's opinion at this present moment, that an hour hence he will be persuaded there is both an external and internal world; and going upon that supposition, I intend to examine some general systems both ancient and modern, which have been propos'd of both, before I proceed to a more particular enquiry concerning our impressions. This will not, perhaps, in the end be found foreign to our present purpose.