IMHO you don't have to use a cold plate - it won't be very efficient, but it won't hurt anything. The purpose of the cold plate is to act as a heat spreader (as Ben said) which couples the relatively large surface area of the TEC cold side to the much smaller surface (NB???) you are trying to cool. Any of the TEC cold side that is not in direct contact won't be able to transfer heat. It will have to be well insulated because it's exposed and will get cold = condensation.
The cold plate also acts as a thermal buffer - to help moderate temperature changes.
On the other hand, the hot side of the TEC "HAS" to 100% be in contact with a heatsink or it can overheat the couples and burn out.
(I won't ask why...
) Good luck