git - pushing branch descriptions to remote

This seems like a fairly recent git feature and may not be well supported yet. Take a look at the README feature, Branch descriptions in git

Considering the description is stored in the config file (here, the local one, within your Git repo), then, no, branch descriptions aren't pushed.

Config files are not pushed (ever). See "Is it possible to clone git config from remote location?"

Simple text files are, though, as my initial answer for branch description recommended at the time.

Branch descriptions are all about helping make an helpful message for publishing.
Not for copying that message over the other repos which won't have to publish the same information/commits.

Using branch.$name.description as the configuration key, give users a place to write about what the purpose of the branch is and things like that, so that various subsystems, e.g. "push -s", "request-pull", and "format-patch --cover-letter", can later be taught to use this information.

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Update 2020 (8 years later):

philb mentions in the comments the issue gitgitgadget/git 438 about "Branch descriptions should be versionable".
philb adds:

If this is ever implemented, branch descriptions would be stored in refs and could then be pushed to a remote.