The Sharpshooter, or "Sharp", specializes in long-range weaponry and headshots. OK, I stole that line directly from the wiki, I admit it.
As I've mentioned, this is a good perk to start any game with because he gets a headshot bonus with the pistol and because he's the only perk that can level using the pistol. He also gets extra headshot bonuses on the sniper rifles and the crossbow. A high-level Sharp with a crossbow at normal difficulties can single-shot a Scrake and two-shot a Fleshpound. His weapons are the pistol, the Handcannon, the lever-action rifle, the M14EBR sniper rifle, and the compound crossbow.
The 9mm pistol (Beretta M9A1) is one of the two basic weapons that every class spawns with. It's a semi-automatic handgun with a medium rate of fire, decent magazine size and cheap ammo. Unlike most weapons it cannot be dropped during normal play -- unless you have two, in which case you can drop one. The iron sights are very basic but it's a fairly accurate weapon in close-to-medium range. Not unlike with the Commando's Bullpup, I recommend that you carefully line up your shots and pick off the specimens at medium range. It's surprisingly effective.
The only way to drop a single 9mm pistol is to die while using it. Unlike the knife, however, another player can pick it up and carry two. Having two pistols will not increase your ammo capacity, however, and while your rate of fire will go up you will lose accuracy (and you won't be able to use the iron sights). On the other hand dual 9mms make a decent backup weapon for in-your-face emergency defense, so pick what works for you. If you find yourself needing the accuracy more than the rate-of-fire you can always drop one and pick it up again later.
The 9mm comes with a flashlight that can be toggled with alt-fire. Dual 9mms are one of the few weapons that a Sharpshooter can carry along side the Crossbow.
The Handcannon, on the other hand, is a Desert Eagle .50 AE (50 caliber) handgun. It has a much slower rate of fire, more expensive ammo, and a smaller clip than the 9mm. It is also more powerful, being able to reliably single-shot a crawler and can penetrate multiple clots with a single headshot.
Like the 9mm you can carry dual handcannons. Also like the 9mm your rate-of-fire goes up and your accuracy goes down, with no change in capacity. In my experience it's worse with the Handcannon because of its higher recoil. It is also -- single or dual -- one of a few weapons that the Sharpshooter can carry along side a crossbow.
Between the choices above, in descending order of preference, I would carry: a single Handcannon; dual 9mms, dual Handcannons; or a single 9mm. I rarely carry dual Handcannons unless I pick one up at random, or happen to notice a new player join between waves who doesn't have time to visit the Trader. (So I'll buy an extra and drop it for him when I see him.) Dual 9mms are a balance between accuracy and killing power, and if I have dual Handcannons, well, I can always drop one. A single 9mm is default and is still useful.
The lever-action rifle is a semi-automatic rifle (Winchester Model 94). It has a decent rate of fire, decent damage, and is the cheapest primary weapon in KF. It's fairly easy to make headshots at medium-to-long range and the iron sights are easy to use. A high-level Sharp can take out a Scrake with only a few headshots (3 to 5 rounds depending on difficulty). Unfortunately you can't carry one along side a Crossbow or it would be my first choice, hands down. Reload is fairly slow, one round at a time, but unlike a magazine reload it can be interrupted partway through. Like the shotgun, you should reload obsessively every chance you get. Reload time gets a boost from the Sharpshooter perk.
The M14EBR is a high-accuracy, high-firing-rate semi-automatic sniper rifle with a laser sight (enable using alt-fire). It is less powerful per shot than the lever-action but has a higher damage-per-second and a magazine reload.
I personally dislike the sight on this weapon, I feel it obscures too much of the target. I prefer the lever-action rifle and tend to use the M14 only when faced with a high volume of specimens, or when I run out of lever-action ammo. I also shoot from the hip a lot.
The compound crossbow is a very high-accuracy, high-powered, low-speed weapon for the Sharpshooter, though it's an amazing weapon for any perk. On higher difficulties the off-perk multiplier drops considerably but is still worth having. It has the second-most expensive ammo of any weapon (only the L.A.W. rockets cost more) and the total capacity is 35 bolts. As mentioned before, a high-level Sharp can single-shot a Scrake and two-shot a Fleshpound, though it depends on the difficulty level (and it helps considerably to have a clear shot at an unraged specimen). It's also a very heavy weapon; the Sharpshooter can only carry dual 9mms, single or dual Handcannon, SCARMK17, MAC10, MP7M or the M79 Grenade launcher. That seems like a large list until you realize that most of them are very expensive off-perk, the MP7M and MAC10 have low capacity, the MAC10 is cheap but has high recoil, and of that list only the pistols get the Sharpshooter's headshot bonus. You could also carry one of the Berserker's blade weapons (Katana, Fire Axe, or Machete).
One thing to note is that a high-level Support Specialist, with his extra carrying capacity, can usually carry a Crossbow plus most any other assault weapon.