Python won't run client-side in the browser, so any server APIs exposed in Python would likely be more in the automation/scripting/admin side of things.
There's no browser support for a python interpreter.
I believe you can already use python on the server side (via geoprocessing)
You could use python as a client to ArcGIS server. You would just be hitting REST endpoints or maybe shudder SOAP endpoints. For instance, I've used python to scrape features from an ArcGIS service via a REST endpoint.
You can use Python Api in ArcGIS server, automating various tasks, like reconcile/post of vresions, registering versioned layers, etc.
but you'll need a license to use it and ArcMap. At least in 9.3.1.
You can get IronPython (http://ironpython.net/) running in the client via Gestalt (http://ironpython.net/browser/) and (http://visitmix.com/labs/gestalt/), and at ArcGIS 93 use IronClad (http://code.google.com/p/ironclad/) to access the ArcGIS .pyd, but then this is IronPython rather than CPython, so it really comes back to Esri supporting IronPython or Esri supporting the IronClad project.