So I just got the basic set and have played a few games with a friend. One thing I've noticed is that so far the games we've done have ended up being pretty one sided...either I stomp him pretty handily or he stomps me. I'm wondering if that's been an experience typical of using the basic set that tends to even out when people have customized their decks with expansions and whatnot, or have folks found it's fairly common for one player to get a big lead very early in the game?

Using just the starting decks, it can be hard to dig yourself out of an early hole. In general though, playing out of a hole takes a bit of skill that really just comes with experience.

as dbmeboy said, the consistency with the starter decks isn't the best because there is only single copies of each set which can make it tough to come back if your down and need a particular or 2 to help turn the tides.

the game can be very swingy and really, one mistake can turn on you and next thing you know your board is clear and/or locked down and all of a sudden your on the defensive. there is always going to be those matches where your opponent just out draws you and you can't seem to fight your way back but I think that is indicative of most card games that require you to draw cards. at least in star wars, you can very easily pitch your entire hand in the edge and draw 6+ New cards.

So do most folks end up buying multiple copies of the base set to have multiples of the same card, or do you get enough copies just through getting the expansions?

imo, you should certainly grab 2 core sets. there is many sets in there that are so good and will likely be used for years, so only getting able to use, say, 1 emperor Palpatine, or 1 Vader or Han, or Luke is just hurting the consistency of the deck.

the force packs all come with 2 copies of each set for deck building purposes so only a single of each is needed.

having said all that, If you just play for fun with friends you could get away with the single core unless they buy 2 core sets. if you want to take deck building and tournaments a little more seriously, grab the second core set.