No. It's all a preference thing. There are plenty of people who do great in fighting games with regular old PS2 pads and the like.
However if you feel like you have a problem performing certain motions or doing other things on a regular controller, then I would totally recommend trying out an arcade stick. Don't buy thing thing thinking you are immediately going to become godlike and start pulling off crazy ass shit though. That takes practice, with or without an arcade stick.
Edit- Here are some well regarded sticks:
Madcatz TE stick - $149.99
Madcatz SE stick - I'd recommend going to the local game stores by you for this one and see if you can find one for cheap. I know a dude who picked one up for 30 bucks during a sale at the mom and pops by his place. Warning: These are have lower quality parts than the TE and I've seen some people complain about how some of their buttons stop responding after a couple months of use.
Eightarc - $169.99
Dualmodded Eightarc(works on both PS3 and 360) - $184.99
Hori Real Arcade Pro.VX SA - $139.99
My friend highly recommends the Real Arcade Pro 3 if you can find it for a good price.
Qanba Q4RAF(works with both 360 and PS3) - $149.99