I have a question about a PC trying to track down where they might find a specific magic item.

I know Xanathar's has a section on downtime regarding research, how research can net you a certain amount of lore.

But how does that lore translate into pinpointing the location of the item?  There will, of course be a quest of some kind, and I'm thinking the item in question may be in the hands of some obscure, but powerful recluse (they want to find an Animated Shield).  I'm not fully sure if I'll make it a heist, a social negotiation, or if they'll just try to kill for it.

I also have some ideas on how one place of research may chain to another, etc.  But it comes down to just how much lore should net the final location?

Is there a formula, say 10 lore for a common, 20 for an uncommon, 40 for a rare, 80 for a very rare and 160 for a legendary?  And for artifacts?

Or do you just wing it and say hey....this is enough.   For the last one, do you do that for all players?  How do you keep it "fair" or at least not make it look like you're favoring someone?

You just wing it. Make it as difficult as you need it to be for the story. With something like this, where they are going after a personal goal, I'd like to have them make a choice. They find out where it is, but its two week's travel out of the way, do they want to allow the bad guy's plans to advance by a month? You can even be sure they understand what the consequences will be, so they can weigh it properly. Or when they get there, they have to give something up in exchange for it, but the thing to give up is plot critical. Something to make it more than just a fetch quest.

Seconding Xalthu's advice.

I sprinkle mentions and lore and hints and several artifacts around my setting.  Something like that shield would be attached to the story of a protector who vanished.

They would follow a series of clues, looking fo the next bit and piece of their lore as part of an adventure.

Then they come across a vast forgotten battlefield, and there upon a massive mountain of the dead, who strangely haven't rotted away, is the corpse of the long-ago hero who had that shield.

Too bad he died fighting and became one of the mindless undead of the battlefield.  You should roll for initiative now. Just in case.

I like to sprinkle bits of this lore and the various formulae for different magic items in the Spellbooks the party appropriates from nefarious Wizard NPCs. Like this: (https://www.dndbeyond.com/magic-items/4743307-markdules-stolen-spellbook).

very creative!