Shortest answer: If you have defined a proper model, you know that R(w1, w2) if in your model the pair (w1, w2) is in the extension of R. If you do not have a model you may be able to infer this from the presumed properties of R, for example from R(w0, w1) and R(w0, w2) you can infer R(w1, w2) if R is Euclidean. You define this relation in the way that characterizes the modality in question. You need to have an idea what properties the respective modality has in the first place.
Longer answer: The accessibility relation has no meaning outside the formal system in general, because its interpretation depends on the purpose for which you use the modal logic. For example, for rational belief sometimes system KD45 is used, which has a serial, transitive, and Euclidean accessibility relation. So if R(w1, w2), then some people would interpret this as "w2 is compatible with what the agent believes in w1". However, from a formal point of view this interpretation is insignificant, because what counts are the formal properties of the relation, i.e. in this case that it is serial, transitive, and Euclidean. These properties directly correspond to axioms in the proof theory of the logic. In case of KD45, for example, they correspond to (D) consistency, and (4) positive and (5) negative introspection axioms. (Axiom (K) holds by default in all normal modal logics.)
I'm sure somebody has written something about it but personally I wouldn't read too much into accessibility relations . They are more or less a technicality, although an important one.
Original, detailed answer: First of all, Priest and Lewis have different views on the question what objects can 'inhabit' different worlds. Lewis is a modal realist, he subscribes to the view that one and the same object cannot reside within two worlds at the same time, which is why he developed counterpart theory. Priest comes from a more modern tradition not based on counterpart theory. As far as I know, he allows in his systems the 'same' object to exist in different worlds where it might have different properties. (He sometimes allows much more, of course, because many of his systems are paraconsistent. Notice that sameness cannot be Leibnizian identity across worlds in this context. Cross-world identity has been debated extensively since the 70s of last century.)
Another question to take into account is whether a constant in world w can refer to an object residing in world u. I don't know whether Priest allows this, but for example Fitting & Mendelsohn do so in their book on first-order modal logic. In case of doubt, you always have check the rules for term evaluation in the logical system.
Regarding accessibility between worlds, both Lewis and Priest need to allow a binary accessibility relation between worlds for technical reasons, as long as they want to stay within the realm of normal modal logic with Kripke frames. If you leave out accessibility, you end up with a system like S5. (I say "like S5", because strictly speaking the system you end up with is S5 with global box modality or S5 with models in which all inaccessible worlds have been removed.)
There are other systems, for example modal logic with neighbourhood semantics, which generalize normal modal logic and do not have an accessibility relation. If I'm not mistaken Lewis studied some of them, too, in his work on conditional logic. However, even in these more general systems you need to restrict the sets of worlds modal operators run over somehow, for example by a 'selection function', in order to make them behave in any interesting way.
A lot has been written about what all of this means from a metaphysical point of view and no final agreement has ever been reached, for it depends on various stances one might take. For example, it makes a difference whether you're a modal realist or believe that possible worlds are ontologically reducible and, if so, in which way, and whether possible worlds are essentially models of a non-modal base language (Hintikka) or not. Cocchiarella (1989, 2007) has worked a lot on the metaphysics of modality, and you will also find many metaphysical aspects addressed by Kit Fine. These authors are not easy to read, though, and require some good technical background.
Addition: I forgot to mention that I know of no author who claims that possible worlds could causally interact with each other, but accessibility does not imply causal interaction.