You are probably referring to the AT Thru Hikers- people hiking from Georgia to Maine and Vice Versa. Obviously they`ve been essentially living on the trail for a couple months so their hair will most likely be grown out they usually have beards and yes, you can smell them coming. They probably cant smell themselves and they most likely bath in rivers and lakes and rinse their clothing all the same. Many times they try to go light as possible so a bar of soap isn't on top of the list for items to bring.

However, for someone looking to do a week-long trip or even a little longer personal hygiene shouldn't be an issue. A little Dr. Bonners soap and you`ll have laundry detergent, dish detergent, shampoo, body wash, and toothpaste all in 1. Your hygiene shouldn't suffer for a short term trip.

Edit:
Yea very good point Will D about the difference between dirty and unsanitary. In the scouts we refer to that as "clean dirt", kind of funny but true. If you drop a piece of chicken on the floor of a restaurant would you eat it- i wouldnt but if I were to drop a piece of chicken into the coals of the fire or onto the ground i wouldnt think twice about brushing it off and eating it. A little dirt never hurt anyone, its the bugs you cant see that you have to watch out for.
References :
NY Backpacking Guide
Eagle Scout

On the trail, you might want to carry a bandana and a small bottle of biodegradable soap to give yourself an occasional sponge bath of sorts – at least your face, underarms, groin, buttocks and feet. This reduces chafing, odors, and bacteria, and you’ll sleep better if not so sticky everywhere. For minimal impact on the environment, carry water away from the source to take this bath if you are using soap – one way is to include in your pack a container cut from a 1-gallon jug. Weighs about an ounce, and can be used for dishes or the sponge bath.
References :
http://www.lowergear.com/product-support/aboutsanitation.html
http://www.trailquest.net/thruhike.html

If you are a clean freak, backpacking for more than a few days probably isn't for you. After the third or fourth day on the trail you pick up a funk. You can't smell it but non-hikers sure do.

Packs and shoes are always the thing that smell the most. Clothes can be washed on zero-days in towns or in the field.

"Horror Stories?" That's all relative. If muddy boots, dirty legs, sweaty clothes, a funky pack, and eating some occasional dirt in your food is unsanitary in you mind, then I suppose it's a Horror Story.

However, since most cooked food requires boiling water for the preparation, you can easily sanitize your eating and cooking utensils prior to dinner. Water filters and purification chemicals will kill or remove almost any microbes found in the water sources along the AT. Hot water can be used to clean yourself whenever you wish. As others have mentioned, carry a few bandannas and some biodegradable soap. (I too usually use Dr. Bronner's for both personal clean-up and washing my cooking gear.)

In my opinion there is a big difference between "dirty" and "unsanitary".
References :

i think the worst part is when you get into the northeast/new england area of the AT, especially around new hampshire and maine…summertime in that area gets daily rain and thunderstorms from all the humidity– you're always packing away wet clothes, tents, etc and never really get dry and clean
References :

Try hanging out with students from the Evergreen State College in Olympia, WA…I would MUCH rather hang with AT through hikers.

Dirty hippies.

To answer the question though:

It's relative…I usually carry a small towel, biodegradable soap, and hand sanitizer with me. That way I can clean up after myself. Of course, I do NOT carry deodorant, anti-perspirant, or shaving stuff. I don't want the extra weight.

I did know people who shaved everyday in Antarctica, however…they did not like their beards frosting up.
References :

when your walking that long i dont think you care . personally i dont care much when im out.
References :