Do you allow third party material in your games?

I would allow because I don't respect the canon and I like to create mash-up mixing things from different franchises. About classes I would test it as boss nPCs with levels.

Some third parties help game with their crazy new ideas about PC races, classes or monsters. Some settings has to be remembered for years.

Absolutely, yes. There’s some incredible third party stuff out there.

In games that I am running, I typically allow third party material, so long as I have access to it, and everyone agrees after some discussion.

I use 3rd party material all of the time in my games - mostly in the form of adventures and monsters (though even there I tend to tweak them to my own purposes).

OTOH - currently my players have no desire to use 3rd party material. My 13th Age group is perfectly happy to use just the core book for everything, and the groups of kids that I run for don't have a lot of access to 3rd party material (and, honesty, are having so much fun coming up with ideas for PCs from the core that I haven't needed to offer any up). So on the player side even though I'm find with it and would absolutely allow it, I haven't had to deal with 3rd party material questions for more than a decade at this point.

The best way to describe my tables is that I only allow second party material in my game, in that my tables tend to be so heavily house ruled that there is little point in using any documents but my house rules.

In general, I make no distinction between first party material (published by WotC for D&D example) and third party material (published under the OGL by Green Ronin for example). What matters to me is whether I find the material the superior implementation of the idea, or which material I find more inspiring. I have no idea really of 'canon'. The fact that it appears in a WotC book in no way means that it is favored. And in any event, there is a good chance that it will be heavily massaged by me before it arrives in the house rules.

Two examples might explain this well - Green Ronin's 'Holy Warrior' class and Green Ronin's 'Shaman' class.

Green Ronin's 'Holy Warrior' class is an attempt to fix the most salient problem with the 3.0e Paladin; namely, that the RAW WotC paladin is only a suitable champion for Deities that emphasize Justice and Righteousness as the key virtues which are laudable in a hero. Deities of a different character or moral philosophy we would expect to have champions of a very different character. This is a very old problem in D&D, as classes like 'The Elven Cavalier' and 'The Anti-Paladin' from 1e and 'The Blackgaurd' from 3e will demonstrate. However, from my perspective the 'Holy Warrior' as presented by Green Ronin still had some severe limitations: it required a specific write up for each deity, there was a lack of balance between concepts, some of the rules were klunky in application, and the class remained heavily front loaded. To resolve these problems, I wrote a homebrew class called 'The Champion' which is not as front loaded, is flexible enough to accommodate any deity without a need to generate custom content, and is IMO better balanced and less clunky. Is this '3rd party content'? Well, if this were an essay I was writing in college, I'd need to cite the original author to avoid plagerization, but I can sincerely state that not a single sentence remains from the original class write up.

Likewise, I really wanted to do something about the RAW Druid for similar reasons. The RAW Druid was tightly bound by flavor to Northern European myths and wasn't in my opinion either flexible or balanced. It was a tier 1 class that could outperform the fighter in combat at most levels, and at the same time compete with both the wizard as a general caster and with the cleric as a healer. That the Druid wasn't flexible in flavor was also something that was obvious from the proliferation of 'Shaman', 'Witch', 'Binder' and other occult magician classes. What I wanted was something much more in the flavor of the Green Ronin shaman, which could emulate much more precisely the concept of an animistic priest whose sorcery was based on appeals to spirits natural and unnatural. The problem was that the writer of the Green Ronin shaman was, for obvious reasons, writing the class to exist alongside the RAW Druid rather than being a replacement for it. Since I wanted the Shaman to replace the Druid completely, it had to do something the Green Ronin writer had pointedly not done, which was kill the Druid and take its stuff. This required a major rewrite of the Shaman's spell list so as to now encompass the Druid spell-list without becoming overly broad and competing directly with the wizard.

So do I allow 'Third Party Material'? I don't allow Paladins, Druids, Rangers, Monks or Barbarians! If you were to pick up a WotC book and say, "I want to take this.", chances are the answer would be "No." If you were to pick up a third party book, the odds of me saying "Yes." wouldn't change measurably. What I probably would do is suggest that the concept for the character could already be played within my rules as they existed, and that if the character could not be played ("I want to be a Were-Orca Minotaur Monk.") that either the lack of inclusion of the concept was deliberate or that if the idea worked for me as well we could work together to create some rules extensions that would support their idea.

It varies wildly.

One of the great things about 3rd edition D&D was the advent of the OGL, making 3rd party content legal as long as it stayed within certain (very broad) limits, unlike old TSR's campaign of actively pursuing legal action against even fan sites providing 3rd party content (much less professional publishers).

However, it produced a VAST amount of material, of a wide variety of quality. Many things were clearly shoveled out with no quality control and only minimal (if any) playtesting. . .but there were other works that were of very high quality.

Ultimately, with 3rd party books, like any 1st party book, allowing something in is on a case-by-case basis on the ideas of does it fit with the game, and is the material of a mechanically high quality.

Just because something is in a 1st party book doesn't mean it's good, or even usable. There have been official 1st party WotC materials that were practically unusable as written (the Truenamer class from Tome of Magic comes to mind), and spells that were insanely brokenly overpowered as originally written (Miasma, which was originally basically a guaranteed no-save death effect against anything that breathes as only a 4th level spell, before it was heavily errataed in the Spell Compendium).

I've known of games that allowed *any* official works of that edition (and known of players that tried to demand games allow books/classes/ect. simply because they were "official" and thought all games must allow all "official" options). . .but I haven't seen that play style in many years.

So, ultimately in any game I there are certain things I allow automatically, which is normally the core rules and books, classes, spells ect. that I know are mechanically sound and will fit in with the intended tone of the campaign. . .and that can possibly include 3rd party materials.

Yes, sometimes.
How do I decide? That old fashioned way - I read it. If I like it & it fits what we're doing, it's allowed. If I don't like it, or it doesn't fit, it's not.
In the case of a player wanting to use some 3p stuff I expect them to A) Provide me a copy/link so I can read it. B) Ask 1st. If they just show up with x & blindside me it gets rejected. C) Respect my decision if the answer is no.

As long as I have vetted it yes.

Yes for stuff I know well. Otherwise, on a case by case basis for stuff I haven’t seen in depth.