Not sure what you mean by "the other side has fallen out." Does a dime fit into a cavity on the reverse? If so, you have a magician's coin that is used to "change" a penny into a dime and sells in novelty stores for a few dollars. If the back is just smooth, you should first check the weight - a jeweler would probably do this for you. If it is significantly less than 3.11 Gm, someone has filed it off. If it is full weight, then you MAY have a fairly uncommon error coin. If two coins enter the dies together, one will get only the "heads" and the other only the "tails". It would have to be examined by an expert to determine for sure.
The note about the weight being significantly different than 3.11 g only holds true for older, non-Zinc pennies. The current penny weighs approximately 2.5 g (I found a range of 2.48 to 2.51 in 5 pennies that I weighed on a balance capable of weighing 0.0001 g).
In addition to the magic trick option, novelties are sold where you can insert another penny and make your own "two-headed" coin.