I cloned to official Fossil repository. Then ran a command to create a new private branch:
<pre>
branch new private_branch trunk --private
New branch:
Autosync:
                Bytes      Cards  Artifacts     Deltas
Sent:            2702         56          0          0
Server says: pull only - not authorized to push
Received:        2473         54          0          0
Total network traffic: 1601 bytes sent, 1523 bytes received
</pre>

Notice the output line: <b>"Server says: pull only - not authorized to push"</b>.

Why does creating a private branch cause a push?