https://docs.python.org/3/library/enum.html#private-names

"""
_Private__names

Private names will be normal attributes in Python 3.10 instead of either an error or a member (depending on if the name ends with an underscore). Using these names in 3.9 will issue a DeprecationWarning.
"""

It isn't clear from this documentation what is meant by "private names".  Please expand on this to be explicit about what name pattern is being described.

I think linking to this part of the docs may help https://docs.python.org/3/reference/expressions.html#atom-identifiers

There's also already a label to reference to, so
the sentence could start with :ref:`Private names <private-name-mangling>`.

Looks good.  Patches welcome.  :-)

_Private__names seems to no longer exist in the Python 3.11 documentation.

https://docs.python.org/3.11/library/enum.html#private-names

It appears to have been removed in this PR https://github.com/python/cpython/pull/23748/files

Should this issue be updated to remove Python 3.11 from the Versions?

It has been moved to the tutorial (links are at the top of the Enum page).

New changeset 7c4d96103c4e16161e9aed9a584c9857d0674099 by Ethan Furman in branch 'main':
bpo-44174: [Enum] add reference to name mangling (GH-29116)
https://github.com/python/cpython/commit/7c4d96103c4e16161e9aed9a584c9857d0674099

New changeset 828722aca4ccba893f6b2e8c1d41fd74fd6e208d by Ethan Furman in branch '3.10':
[3.10] bpo-44174: [Enum] add reference to name mangling (GH-29117)
https://github.com/python/cpython/commit/828722aca4ccba893f6b2e8c1d41fd74fd6e208d

New changeset e628700dbf2c3376502cbb5a9bff2d58d1102e16 by Ethan Furman in branch '3.9':
[3.9] bpo-44174: [Enum] add name-mangling reference (GH-29128)
https://github.com/python/cpython/commit/e628700dbf2c3376502cbb5a9bff2d58d1102e16