hasu, your inquiries seem related to some confusion regarding the operation of a TEC.
TECs (aka Peltier coolers) are, at their very core, composed of a bunch of circuit elements known as thermocouples wired together. Each thermocouple is composed of two parts, and each part of made of a distinct material. That is why you will see thermocouples referred to as being, say, p-type/n-type bismuth telluride or what have you. The reference describes (in brief) the materials used to create the thermocouple. Anyway, when you run voltage/current through the circuit, the thermocouples exhibit what is known as Peltier effect: the transportation of heat through each thermocouple from one side to the other. It is Peltier effect that creates a "hot side" and "cold side".
If you were to try to break a TEC into two parts, you would be effectively breaking apart the thermocouples themselves. Physical contact is required for the two parts of the thermocouple to function. Without that, you get no Peltier effect. So, no, you can not simply take a TEC apart and move the "hot side" and "cold side" into different locations. Maybe someday we'll be able to exploit quantum teleportation effects to facilitate the flow of heat across space and time, but for now, we can not do that.
Examples such as aigomorla's are about as good as you're going to get. Traditionally, TECs have required cooling to keep the hot side temperatures low enough that the cold side can produce desirable results. Water loops are often used for that purpose.