Sony has gone ahead and done it. In what will presumably be their answer to Microsoft’s Game Pass offering, they have confirmed that from now on, you will be able to download PS4 and PS2 games available on the PS Now service to your console and play them locally (in addition to streaming them, if you’re so inclined).
You’ll be able to do this for as long as you maintain a subscription; this will apply to PS2 games that were re-released on PS4 via the PS2 Classics banner, and PS4 games, but not PS3 games (since emulation of PS3 is extremely complex, and presumably not something the PS4 can handle natively)- for those, you will need to stream.
Your save file from streaming will transfer over to your console as long as you have PS Plus, but, more interestingly, you still won’t need PS Plus to play PS Now games online, even if you are playing them locally- which makes this a very appealing proposition to a lot of people, I imagine.
PS Now has a bigger library than Game Pass, so Sony is off to a good start here; now to see if they will eventually start releasing their major releases on PS Now day and date like Microsoft does with Game Pass.