Constriction of the blood vessels in an area exposed to cold is the body's way to preserve heat loss by reducing blood flow, termed vasoconstriction. The extremities are particularly vulnerable due to their small volume, relatively large surface area, and little metabolic activity to produce heat in the area. This is the desirable effect achieved by icing a swollen body part e.g. sprained ankle. If the ice is left on to long, or the exposure to cold is to long or intense the body will undergo what is sometimes termed a "hunting" reflex or response. This response actually dilates or opens the vessels in the cold area. This response lasts about 5 minutes to provide some protection from cold by bringing blood, therefore heat, to the area. Prolonged, repeated exposure increases this response and offers some degree of acclimatization. This is the main reason we are told only to ice a body part for 20 minutes, as this response is undesirable when treating inflammation. We don't want to increase blood flow to that area. It is, however, the likely reason you are experiencing this "accommodation" to the subsequent exposure to the cold as you are feeding into this reflex by heating your hands and the body will adapt to subsequent exposure to cold by not overly constricting the local blood flow. The practice described by your friends may be effective but possibly on the extreme side. Although it is an extreme example, if exposure to cold is severe, i.e. hypothermia, warming your hands or feet is a big mistake as dilating these blood vessels will send that cold blood to the core of your body, thus reducing your overall body temp further which could be dangerous if not fatal. That is why people pulled from frozen lakes or mountain exposure are warmed only at the body core first and not the extremities. Not that it applies to the cold hands you are experiencing on your bike, the possibility for it not to work exists, leaving you with self-induced frozen hands. I would not think you would cause any arthritic conditions, but this is not to say that you would not exacerbate an existing condition or potentially create one especially in very cold conditions. If temperatures were in the 30's F, I would not try the "experiment". Obviously if you have been diagnosed with Raynauds, you would want to avoid this practice or any exposure to cold. Getting adequate gloves, keeping hands dry, brush guards for wind protection, staying hydrated, avoid access caffeine, and grip warmers sounds like a good place to start if you make a habit out of riding in cold temperatures. I highly recommend a pair on "Cool Max" material glove liners. Wear them inside your normal gloves. I use them skiing, running and riding as they "wick" away moisture from your hands providing better warmth. They are also very thin, not to interfere with activity.
Good Luck, Keith