To transfer the dog from one video to the other, you will have to isolate the dog from the background. This can be done ‘easily’ using the bluescreen method.
Rotoscoping means that you remove the backgound frame for frame, by painting over the background. This can be done in the latest version of Photoshop or in e.g. After Effect, but is very labour intensive.
Taking your dog to the beach will be a much better/faster solution.
The alternative idea I came up with is the BCC MotionKeyUnit. This plugin for AfterEffects removes moving objects from a scene based on a frame to frame comparison of all the frames. If this plugin has a kind of “reverse mask” option, you could use it to isolate the dog. But anyhow, to use this within Vegas, you need an AE-adaptor like Boris RED, Boris FX or After Effects itself.
Theo