There's been discussion on this every so often. No, a cold plate chiller or coil chiller(for soda or beer) needs to be set in ice but not actually freezing(more explanation to this).
Putting the plate chiller in a fridge alone will not be cold enough to flash chill the soda water (or whatever liquid is flowing through).
Putting the plate chiller in a freezer will wind up being too cold, and you'll wind up having water freeze in the plate itself.
Ice on top of a chill plate (ice bank) will: (with the ice being below freezing) keep the surrounding liquid at a constant freezing temperature (32F)without being so cold as to freeze the liquid in the line (31F). If you kept this in an open bin, you would lose ice due to the surrounding ambient temperature, so it is said to be possible to have ice in chill plate assembly that has a cooling system that doesnt' get to freezing. (like having a chillplate in a box in the fridge that you keep adding ice to. You'll still be losing ice, but some of the ice won't be melting as quickly). Really though, the only ice that's lost this way is the outermost ice (like the stuff sitting on top exposed to air).
See:
Beer Jockey Box | Beer Cooling & Dispensing Equipment
A chiller assembly is used when dispensing soda or beer. For example, if you have an inline system (no kegs) you're taking water right from tap which is approx 50 degrees(I'll measure later) and either chilled first then force carbonated or more likely force carbonated then in-line chilled to 32 degrees, mixed with soda syrup (especially for a bag in box method) and then sent out through the soda machine / bar gun.
You'll usually see this type of usage at a bar or restauraunt.
An alternative to the chill plate method is to force carb the water and send it into a keg that's stored in a fridge. The 5 gallons of water will chill to 32 degrees and be distributed out, and the carbonation system will replenish this water. 12 oz water at 50 degrees adding to 5 gallons (625 oz) at 32 degrees means that your temperature change will be negligable. However, this won't stand up to constant use (like the hundreds to thousands of customers at a bar or restaurant) but would stand up to some home use. Maybe even a party of guests who don't constantly drink soda. Check out some of EFaden's posts where this setup was used.