Barking dogs and sound isolation

It is not always "symmetric" (technically, this property is called reciprocity)!

However, assuming the medium between the two dogs is linear and microscopically reversible, your intuition is correct. Under these assumptions, it is actually a theorem, due to Lord Rayleigh, that if you exchange the source of the sound and the point at which you measure it, the sound level is unchanged. (“ Some general theorems relating to vibrations,” Proc. London Math. Soc. 4, 357–368. https://doi.org/10.1112/plms/s1-4.1.357). In electromagnetism, this is known as Lorentz reciprocity.

Now if we break one of these assumptions, we can avoid the consequences. For instance, imagine there is a very strong wind going westward, going faster than the speed of sound. Then, barks emitted by the west dog will never reach the east dog, while the opposite is possible. Such a system is called non-reciprocal.

This is an extreme, but straightforward example. You can get around reciprocity in subtler ways. Read https://science.sciencemag.org/content/343/6170/516 for a recent reference on the subject.