All the attenuated live viral vaccines spread. This is one of the reasons they are so effective since they keep herd immunity up. For example, every kid that gets the live polio vaccine spreads it around to their peers and parents.
Can you point me to sources supporting that claim? I searched myself, and what I found says that our knowledge is limited to the polio vaccine:
"Little attention has been given to vaccine transmission, possibly because transmission is rarely measured and largely unknown in humans except for the oral polio vaccine. Whether transmission is indeed rare for other live vaccines, or has merely gone unnoticed, is not clear – polio vaccine transmission is accompanied by evolution to high virulence, creating problems that draw attention to transmission."
From "Transmissible Viral Vaccines", James J. Bull et al., Trends in Microbiology, January 2018. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5777272/
It doesn’t appear that horizontal transfer is absurd, and while I’m not immediately seeing a large body of evidence that it happens all the time, it does seem to be generally accepted that it occurs.
Here’s what I believe to be an informative article on the topic: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3667938/
The polio vaccine actually went to the inactive form to reduce the risk of polio from one of the vaccine strains reverting to wild type and infecting others. It is still less effective at achieving herd immunity than the live version because of the lack of viral spread.