AFAIK, neither company has authorized hobby development on either console.
The steps for becoming a registered developer allow for "indie" development, typically referring to small but established teams that are independent -- that is, the studios are not under contract with existing publishers.
Microsoft stated that they intended to support small hobby developers again with the XBox One, and that it may take some time. It was several years after the launch of the XBox 360 that XNA was released, and we are still under a year out from the XBox One launch.
XNA was originally a wrapper for cross-platform DirectX 9 era development. That did not change. The system targeted systems that were frozen, and it was not updated and maintained to current versions. If you want to do DirectX 9 development on Windows and sign up for the Creator's Club and do XBox 360 development, or develop for a now-defunct version of Windows Phone, XNA will still accomplish those goals. The targets remain the same, and the software is still functional.
For current generation consoles, it appears so far only the WiiU has been exploited to run hobby software. I don't see anything in the homebrew communities about either PS4 or XB1 for running unsigned code or signing code to enable it to run. (Although there do seem to be the normal problem of unlawful disc images.)
Home development on the consoles is rather difficult, with the notable exception of Xbox 360 through XNA. Before that you need to go back to the 6th generation and earlier to find systems that a single hobby developer can produce quality stuff. In the 6th console generation both Game Boy Advance and Dreamcast systems are still popular in the hobby community. There are also popular Atari 2600 (2nd generation console) hobby sites. Some Super Nintendo (4th generation) and NES (3rd generation) homebrew groups are out there, but are less popular. I've run across some rather fun stuff on the Vectrex as well, a former co-worker was a collector who loved that chip, and even wrote several Vectrex simulators for various devices. The older consoles have emulators so you can play on your smart phone, which makes them rather fun. Climber 5 and Marble Craze are two examples of excellent homebrew games on old systems.