My hot water heater is now leaking. I'm going to install a new one and it appears that the builders located the expansion tank on the output hot side 8 years ago when the house was built. I'm to understand it should be on the cold side. What would be the effect of it on the hot side? Thank you in advance collective intelligence!

Should be on the cold side, not the hot.
Pressure changes on the hot side pushes back to cold.

To the best of my recollections, there is no check valve on a water heater. The tank could go on either side. Personally, I'd want it to dangle vertically from a well-supported ¾" sweat x ½" female tee (mostly copper in these areas) in an easily-accessed location. I'd base the installation on maintenance and replacement criteria.

On the other hand, the one water heater expansion tank installation PDF I consulted recommended installation on the cold side. Further, although their tank is rated for both hot and cold water, the tank is not recommended for continuous use above 140˚.

So I guess I was a whole lot of help.