Recently I have been experiencing An odd noise from underneath my house and a few weeks back I had an issue with a neighbor who voluntarily wanted to flush his toilet and have me listen to see if I could hear it.

Is it a condo ? Apartment ? Single family dwelling ? Is your neighbor above you, on upper level ?

Your toilet and neighbor's toilet are draining into the same pipe. Yes, you may hear the noise, especially with ABS drains and / or pressure flush toilets installed

Milo