Just wanted to chime in and say I'm having this problem also. I noticed when trying to save some pictures from some text messages that friends sent me - they were going to the "Recents" album on my iPhone, but not the Photo Stream album. Same for pictures taken directly from my iPhone camera. Like some others have noted, I did notice pictures saved from a website, or screen shots, were actually making it to the Photo Stream.
Did the usual process of getting Photo Stream going again: Switched off My Photo Stream on my iMac and iPhone, signed out of iCloud on both, rebooted both, signed back in, switched everything back on - and same issue. Oh boy...
Very annoying. As others have mentioned, My Photo Stream has been getting worse and worse over the last couple/few years. I'm with those that believe they are letting it die a painful death to push us all to iCloud Photos and the associated extra storage costs.
I'm not against using iCloud Photos in theory, but I don't like being forced from something that worked great for me to something I don't really need - and being made to pay for it to boot.
I realize I could change the format in my camera options (and maybe have that fix it for who knows how long), but I'm tempted to just switch off Photo Stream altogether and use Air Drop or even the flippin' lightning cable to manually sync my photos every so often.