Given the narrative about the Tories breaking the system, it's probably the best chance they'd get to do something like this. However there are also a lot of left wing news sites, whose main pleasure in a Labour win is that they no longer have to think about the Tories, and can now concentrate on the real enemy.
Streeting has certainly talked about using the private sector, but as far as I'm aware has been very clear that they won't be charging for NHS services.
Still, there was a post on here recently linking to the last Labour Government introducing a two-tier NHS, where you got a second-tier service if you didn't pay. The actual article was a mention in a speech about the NHS branching into areas that it had never covered, and using its expertise/economies of scale to plough money back into the more important work. The main people complaining about the suggestion were of course private providers.
I'd be interested if it's the same kind of source, or something more concerning.