The only way for coolant to be let into the exhaust system is if the head gasket is leaking.
The probable cause of coolant loss will be down to the expansion tank (the bit you look at to see if it needs any adding).
Basically on a cold engine the level is fine, once the engine is started only part of the coolant is being circulated as just the coolant in the block and maybe the heater matrix is keeping the block cool.
As the temp rises in the coolant it is in contact with the thermostat, once it reaches a certain temp it will activate the thermostat and allow the coolant to flow round the full system bringing the cold coolant from the radiator and pipes to flow around the engine block which would then also drain coolant from the expansion tank as well.
As the coolant heats up it expands and it expands into the expansion tank. If it heats up too much it will fill the expansion tank and if it needs to put more into the already full tank it will force out the excess out.
You might not see it as the engine bay will be hot and will vaporise the excess and may leave no trace.
Another reason for the expansion tank to fill to "over - flowing" is the speed of the engine.
The faster the engine turns the water pump the faster the coolant flows, if it expands into the tank and then over flows and then you slow or stop the coolant will retreat back into the cooling system and when you look the coolant level will look fine until it cools and contracts.
Might take some time for it to show a loss depending on the amount it of excess it has to get rid off.
