I've never liked the Complete Priest; it felt far more like it was about designing a priesthood than about playing a priest, and the priests produced were laughably weak compared to the cleric, or even druid. It mostly just disappeared from my memory, partially because I always felt it was so bad. But, it also means I haven't examined it in 30+ years, so I'll take a look now.
Amazon Priestess: Like the Complete Fighter Amazon, this is pretty lackluster; a +3 to hit and damage on the first blow against sufficiently bigoted enemies, and a -3 reaction adjustment with the same. A couple bonus NWP. None of this really improves their role as priestess, though, as with the amazon fighter, it at least flows out of the early 90s tradition of "women get to be warriors even if men don't think they can be." Which, given 1e strength limits on women, might make some sense, granted, but it rings very hollow now, and that it doesn't support them as priests is even worse, IMO.
Barbarian/Berserker Priest: This is an interesting one, not because it is, itself, cool, but because it synergistically improves the Berserker from the Complete Fighter. You get a bonus NWP, a modifier to reactions (positive with barbarians and berserker peoples, negative with non)... but you can also make berserkers go berserk more quickly. Normal? 10 rounds. With a priest nearby? 5 rounds. If the priest is capable of inspiring a berserker rage, they can do it in one round, and you don't even need to use the power. It creates some layers to character creation, but if you don't HAVE a berserker in your game, it's not much.
Fighting Monk: Another interesting one, though it also illustrates the weakness of priests created with this book, and why we never used it. Free Tumbling, can take NWP from any of the five groups without penalty, and gets two free NWP for specialization in unarmed combat. They can't wear armor or have a lot of stuff, which is pretty standard for a monk, but the game also assumes that limiting them to 3 Major spheres and 2 minor spheres is reasonable. Keep in mind, the PH cleric has SIXTEEN spheres of influence... this one is supposed to top out at 5, two of them minor.
And, look, I have ranted about the power of the cleric in the past. They are a powerful class, being 2nd best as both warriors and spellcasters. But the Complete Priest has what I consider an absolutely bonkers approach to this problem, where they try to cut priests back amazingly, with little trade-off. A Fighting Monk gets some neat benefits, and I think making one off a basic cleric is some decent trade offs with the "can't wear armor" and "can't keep stuff" requirements. But this one? Sweet blessed monkey.
Nobleman Priest: Mechanically, bonus NWPs, bonus cash, and reaction modifiers, plus the "can demand housing/must house others on demand", and requirement of conspicuous consumption, like the Complete Fighter nobleman. Their advantages are almost a copy/paste of the fighter kit, but their disadvantages aren't... the fighter's ability to fall into hugely negative reactions by failing to maintain appearances is missing from the priest. It somewhat moves the Nobleman Priest from the fighter's "OK, this is kinda cool" to "Yeah, no." It just isn't as interesting or mechanically based as the fighter it emulates.
Outlaw Priest: Upside? Free Religion Proficiency, no superiors. Downside? Superiors will try to stop you. Mechanically, even socially, it's boring.
Pacifist Priest: TBH, kind of interesting, if not something I necessarily want to play. You get a whopping +2 to Charisma, and +2 to reactions on top of that if someone isn't completely opposed to your pacifist philosophy. Downsides? You can only buy proficiency in bow and dart (why not sling? Or anything else? Who knows), and can only use them in competition. You cannot wear armor or use weapons or you lose your spells for a month. It's interesting, certainly, but even the text says "Yeah, this isn't a good idea for PCs."
Peasant Priest: Peasant priests can get shelter from peasants in their own communities, and get +2 to reactions from all peasants, which nice. Bonus of Agriculture or fishing and Weather Sense or Animal Lore (not animal handling? Or husbandry?) In return, they have some VERY strict limits on ownership. Other than weapons, they can only have one item worth 15gp; nothing worth more. Everything else must be 10gp or less, and the total can't exceed 75gp, again excluding weapons. So, they're barred from almost all armor, and, depending on your reading, might be barred from all magic items as well. It is a heavy load to carry, and I think makes the peasant priest unsuited to most D&D.
Prophet Priest: Gotta say, I kind of like this one. Take a standard priest, give him Weather Sense NWP, and the Prophecy Granted power from earlier in the book, and a -2 to reactions because We Don't Talk About Bruno... with the caveat that the reaction penalty won't make people more than Cautious. It's a nice balance of abilities... you get a granted power your priesthood doesn't normally grant but reasonably could, and it makes people nervous. Mind, the granted power is itself wishy-washy as all get-out, but it's a good balance at the level of the kit. It might have been nice to see this with the Barbarian/Berserker priest, really.
Savage Priest: Bonus Weather Sense or Direction Sense, Endurance or Survival, with a level/day detect magic, and a penalty to reactions from non-Savages. Comparing it to the fighter savage, it's a little meh, because the fighter savage had more options. Unlike the Savage Warrior, he can wear armor and definitely use a shield, and doesn't have the Savage Warrior's limitation on future use of armor, unless the DM takes the "Use the Savage Warrior equipment rules" option given in the kit. And I know I liked the prophet priest and they're much the same thing but... this one feels boring. Like they slapped Savage on it and cribbed from reading part of the Savage Warrior but changed just enough that the professor wouldn't accuse them of plagiarism.
Scholar Priest: This one is, IMO, fun. You get a bonus of Reading/Writing, can turn WP into NWP if you like, get a bonus with most scholars... but have a 1 in 6 chance of any random scholar HATING you. Like, instead of a +3 with this scholar, you get a -6, because you argued with them about something. A negative 6 reaction modifier is as if you had a TWO Charisma. It can be as much as an effective 10 point penalty to Charisma with this one person. It is LOVELY. The person who wrote this kit has clearly talked to people in academia, if not been one themselves.
Overall? Really boring. Of them, I only really liked the Prophet and the Scholar; the Pacifist was interesting, but not something I want to play, or particularly want at my table. I think I'd like the Barbarian/Berserker better if they actually gave you the Incite Beserker Rage power, like the Prophet gives the Prophecy power. But most of them felt like copies, or poor copies, of ones in the Complete Fighters, while the Fighting Monk was more "I decided I was going to completely rewrite the priest class and make you pay for it."