Should I save money or pay off student loans quicker?

What are the rates? I would assume that would be the determining factors.

I think paying off your loans is the best option. Placing money in savings will not do you much good besides building an emergency fund. The loan + interest will cost you more in the long run than by paying the minimum and putting the rest away in savings. If you could get a higher return through an investment, you might could justify paying the minimum and investing the rest.

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I can afford to take about $1000/month ($500/pay period) and either put all of it towards my loan payments, or put half in savings and half towards the loan.

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I think you can be more aggressive than that, but that's just my opinion. True, $4K won't last you that long but the risk isn't high or you wouldn't be asking.

Paying off the loan is a guaranteed 6.7% return on investment tax free. That's a very nice return and worth pursuing.

ETA: As others mentioned you can also play the low APR credit card game, that can be a useful ace up your sleeve. My suggestion is not to transfer anything (those offers tend to come with fees) but just live off the card and pay the minimum until the teaser runs out. Just be sure you can either pay it all off then or roll it over to another competing offer to keep knocking out the higher-rate student debt.

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I have $20k in grad student loans.

I can afford to take about $1000/month ($500/pay period) and either put all of it towards my loan payments, or put half in savings and half towards the loan.

I don't know which would be smarter... any advice?

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Married? Kids? Wife work if so?
If not I'd hit the student loans hard and just get rid of it. 4K is enough emergency fund for a single young person.