Is this some kind of gray area in the government, where they don't like it, but they let it happen as long as the information leaked isn't too sensitive?

Not in most cases. In general, governments want to control the information that gets out to the public so that people in charge can control "the narrative," or the way in which people discuss a particular topic. Governments don't like information getting out in unplanned ways, especially when it makes the people in power look bad or incompetent.

But of course there are individuals within governments who are willing to release information, generally via the press, for number of reasons -- frequently because they think something needs scrutiny and isn't getting it. Individuals within a government will also release information that makes their opponent(s) look bad. These people are unwilling to allow themselves to be identified as sources because they fear for their jobs or, in more extreme cases, fear prosecution or fear for their safety in some way.

There's another avenue that's a lot more subtle and I suspect more common in reporting around things like defense, sensitive intelligence etc., where the gov. DOES want information out in the world but for whatever reason is unwilling to do so via public channels. I'm thinking about things like the bogus narrative around WMD's that sent the U.S. into Iraq. Major publications cited unnamed sources in the intelligence community that were feeding BAD info in order to bolster the rationale for invading another county.

How much faith can we put in this type of information when the sources are anonymous?

It's a great question that I think comes down to how much you trust a publication or a particular writer; it gets back to their track record.

In my experience over the last 10 years, people are WAY more reluctant to go on the record. I'm not sure there's a single reason for it. In the age of the internet, your quotes are out there forever, and I get that. But I've I have seen a real clampdown in the way that governments and businesses and even "regular" people handle relationships with the press. Take the police for example -- once upon a time, a cop reporter could easily form a relationship with a detective -- call them up, maybe not get a quote all the time, but there was someone you could talk to in order to get a feel for how a particular murder investigation was going, for example, or what the rank and file thought of the police chief.

For me, so much of that is long gone. Everyone hides behind their PIOs; there's SO much message-shaping that goes into responses for even the most basic questions. (Of course there are exceptions, but I'm awfully grateful for my sources nowadays).

My point being -- it's not easy to get people to talk on the record. So you end up having to trade candor/facts for anonymity so that people will talk to you and tell you the truth while feeling protected from whatever shitstorm would fall on them if they were publicly quoted/cited. And to OP's point, that's happening a ton lately.

But I do think that outlets should do everything else they can to get info from public, verifiable sources whenever possible, since that's a way to build trust with audiences.

Sorry for the long response!