There are two basic families:

the ones ending in --ol like sorbitol, mannitol, and xylitol, are most commonly found in dietetic candies. Sorbitol is used in prescription compounding to make liquid dosage forms from tablets/capsules. It is a sweetener and laxative. It is used straight as the laxative in hospitals and long term care. Not so common in outpatient pharmacy settings. Mannitol is sometimes used IV in head injuries or to treat malignant hypertension on an emergency basis. It has other uses too:
http://www.rxlist.com/cgi/generic/mannitol_ids.htm

The other "fake" sweeteners are Equal (aspartame), Splenda(sucralose), acesulfame, and saccharine. These typically do not cause diarrhea. However, the powder forms have maltodextrin in them to extend them and this may cause diarrhea in sensitive people.

Many people get headaches from aspartame, and this may be a function of the aspartic acid in it. Glutamate and aspartate are stimulants to the NMDA pain receptors.