My brother and I are looking to get into the Warhammer LCG - for casual (though competitive) play between the two of us, only. ...and are just not sure what is suggested for a solid start.

In looking into the other LCGs (I'm a fan of Lovecraft so I'm also considering checking out the CoC LCG in addition to the Warhammer one), it seems for those LCGs its very common or highly suggested to purchase a minimum of two copies of the Core Set - so that you have a little more wiggle room for deck building (as you only get one copy of many of the good cards in a Core Set, where three are allowed in a deck) and to make single-faction decks a possibility (as there aren't enough cards for each faction in one Core Set to accomodate single-faction, you're forced to mix).

Is this how Warhammer LCG is, as well? ...is it highly suggested you buy at least two Core Sets for a fun and competitive casual play experience (and allowing for single-faction decks)?

Any info/suggestions appreciated.

The Warhmmer LCG is packaged in the same fashion. So you'll most likely want to get two of the core sets.

If you are new to all the LCG games I'd suggest starting out with Warhammer. It just started, the second expansion is just starting to trickle onto shelves and you can get into the game early. The other games have a lot of expansions already available.

I am in the same situation as the OP, only plan to play against family and friends and they will only be using my cards (we each pick a game we like and all share that one persons purchases) and so far we have all been happy with just one copy of Core and Skavenblight.

My suggestion would be to buy just 1 and if you find you are having problems over who gets the neutrals then get a second copy.

I'd also say just go with the one core set - I've had no problems with only one core set, and if I did I honestly would probably just play something else rather than fork over £30 for a few extra copies of certain cards.
Having a shared stack of neutrals is fun, I find. If we're playing unconstructed, we just deal 10 to each player, pick a faction, and mix the neutrals in - as Sigmar intended. Not knowing what you're getting in terms of the neutral element of your deck allows you to work out your strategy on the fly rather than tune your deck like a machine.

Well, I think it's fun anyway.

If you're just playing between the two of you, I recommend just getting one copy of things. Think of it like a boardgame in that regard. You don't buy two copies of Chaos in the Old World so you can have more bloodletters.