There are 2 reasons for this reaction that I am aware of.
Massage stimulates histamine release in the body and one of the effects is that it increses the amount of fluid in the sinuses.
Also, have you ever noticed that when lying in bed, if you are lying on your right side, your right nostril closes and the left one opens up? This is due to gravity and with the additional fluid in the nasal cavity caused by the histamine release, results in a stuffy or even runny nose when lying face down during a massage with or without the face cradle.
As mentioned, use of essential oils like eucalyptus and peppermint on a tissue and either sniffed after the massage or tucked under the face rest cover can prevent or clear it up. If that is not enough, a hot steamy shower or bath after your massage usually will. Because of the detoxifying effect massage has on the body, we recommend that everyone drinks plenty of water following a massage to flush out the toxins. I also recommend a nice hot 15-20 minute bath in epsom salts especially if you tend to be sore afterwards. Both of these will aid in gettign rid of the stuffiness.