depends how cold it is...there is some sensitive equipment inside your shorts. I wear a set of thermal cotton boxer-like shorts under my cycling shorts when the temp gets low...(sometimes two pairs if its really windy.)
You don't need to wear anything under cycling shorts, they are designed that way. Just get a good pair with high quality, soft material and a good chamois and you will be fine.
Spandex (Lycra) cycling shorts have a pad, usually of synthetic chamiox, and are designed to be worn without briefs. However, unless you are able to wash them each time and wear a clean pair every day then it is better to wear something under them.
Most types of underwear are not comfortable to cycle in because of the position of the seams, but there are a few that are OK. Also there are cycle-specific short liners, including some with a windproof section to stop your nuts freezing in cold weather.
Personally I prefer to wear something over the top in cold weather, and have enough pairs of lycra shorts to allow me to wear a clean pair each day without any kind of underwear. A well made pair of black shorts will preserve your modesty.
It does make for some good laughs when someone (always a non-cyclist) is admiring my bike and fondles my saddle--no concept of where that saddle has ridden.