One may object that this latter notion makes no sense: each person should believe whatever is best supported by their evidence, and that's all there is to say about epistemic justification. There is no more sense to be made of 'general justification' than 'general evidence' (unless this simply means 'evidence shared by most people', but there's nothing especially significant about that -- my notion of objective justification is not merely that of widespread justification), or so the objection goes. I think we actually may be able to elucidate an epistemologically significant notion of generally available (in circumstances C) evidence for a proposition, i.e. evidence that an epistemically virtuous inquirer (in C) would quickly acquire, such that any person (in C) in some sense really ought to believe the proposition even though -- due to their negligence in acquiring all relevant evidence -- their personal epistemic situation excuses or even licenses their contrary belief.* But I don't need to make so strong a claim. It suffices for present purposes to restrict our attention to a priori justifiable propositions, and so avoid these complications.
* (Clayton might say that their belief is rational but unjustified.)
The propositions of a priori philosophy and mathematics don't seem to be supported by evidence in the ordinary sense. Compare: our contingent experiences provide evidence for some empirical beliefs over others, and what we ought to believe about the world depends on these contingent observations. But truths of reason seem importantly different. Whatever "evidence" or reasons there are to believe one philosophical claim over its negation don't seem contingent in quite the same way; rather, they are presumably available (broadly speaking) to any rational agent as such -- hence the label 'truths of reason'. This allows us to drop all that complicating talk of "evidence available in circumstances C"; a priori justifications are "generally available" in the strong sense of being universally available (to sufficiently rational agents). We can thus understand talk of a proposition's being 'objectively justified' as simply a matter of its being a priori justifiable.
Importantly, we are not perfectly rational agents, and so our cognitive limitations might prevent us from subjectively grasping the objective justifications for believing some philosophical proposition. This creates a need for "non-ideal" epistemological theory (by analogy to non-ideal moral and political theory). Although the proposition may (ideally) warrant believing in virtue of the existence of an objective justification, we personally - as non-ideal agents - may justifiedly believe its negation, say due to our having a non-culpably incomplete or distorted understanding of the matter. Our non-ideality is revealed by the fact that we don't believe the objectively justified proposition. Nonetheless, our believing of its negation may be epistemically virtuous in its own way -- we may be successfully following epistemic rules (or heuristics of reasoning) that are generally more fit to our limited rational powers, even if they lead us astray in this particular case.
So I think we can make sense of a distinction between 'objective' and 'personal' justification, at least in relation to a priori beliefs. Note that this now opens a new position in logical space -- besides old-fashioned foundationalism and coherentism -- for the hybrid view that objective justification has a foundationalist structure, whereas personal justification has a coherentist structure.
(Aside: I'm inclined to attribute something like this hybrid view to Sidgwick, for reasons detailed here [II].)
P.S. Do epistemologists standardly make this distinction, or something like it? I don't think I've ever seen it explicitly discussed, but if you have, please pass along the reference!