Although I think the other answers do provide good information, I would argue that, as posed on
sustainability.stackexchange.com, all of them are still wrong (albeit maybe Chris Travers' answer is only partially so).
You should not put meat in your compost pile.
Sustainability of Eating Meat in the First Place
If you're even remotely concerned with sustainability (underlying assumption I'm making, based on where this question is), then you shouldn't be eating meat at all. Meat is hands-down the most inefficient, unsustainable way to feed a human being, as it stands today.
To really go back to first principles, consider the laws of thermodynamics. You can't create energy from nothing, and no process is perfectly energy efficient. In order to build an animal from food, first you have to grow, process, and transport the food. The animal has to eat it and process the food into energy, which it does not do with perfect efficiency. Then, the animal has to live for a while (weeks, months, or years, depending on the meat), during which time it has to move around, eat, poop, heat itself and make noise. All these things take energy.
Instead of performing this cycle, skip the animal altogether and just grow (non-animal) food that can be directly consumed by humans.
How inefficient it is to eat meat depends on the animal, what you feed it, and the processes used. There is no one answer here. But, a reasonable range, for example, might be that in order to produce a given unit of beef energy (e.g. a calorie), you have to put in 6 to 50 times that much energy in the process. That puts the effective energy efficiency of the process at between 2% and 16%. That's terrible.
Unfortunately, it gets worse. The previous figures only discussed energy efficiency. If you believe that climate change is a problem, you need understand that not all greenhouse gases are equally potent. Methane is 22x more effective at trapping heat in the atmosphere than is CO2, and the animals we get our meat from produce a lot of methane, meaning that their contribution to global warming is even worse than suggested by looking at their energy efficiency alone.
So, don't eat meat (cow, pig, sheep, deer, etc.) in the first place. Poultry (chicken, turkey) is a bit better. Seafood is better still. A vegetarian or vegan diet is most sustainable.
If You Do Eat Meat, Don't Compost It
As has been said many times on this site, including in Chris's answer, sustainability choices need to be prioritized to reduce use first, then reuse second, then recycle last.
Why do you have extra meat at all? If you have leftover meat, then buy a smaller cut. Virtually everywhere I see meat sold allows you to buy pieces in an almost continuous array of sizes.
In the developing world, currently about 1/3 of food is lost, mostly to disease or spoilage (lack of preservatives, but also refrigeration). However, in the developed world, the loss is almost exactly the same (1/3), for different reasons. One reason is people buying and preparing more food than they need and just throwing the rest out (or composting it). If you need to eat 2000 calories per day, then prepare 2000 calories. Don't prepare 2500, and then compost the unused portion.
If you find yourself in a scenario where you can't find a way to stop eating meat, try to buy the correct portion, but you still have leftovers, then save the edible meat for tomorrow. Chop it up to make a lunch sandwich. If you still have fat and gristle left, those things probably won't break down quickly in your compost bin, and can create health/rodent concerns outlined in other answers. Just bury the fat in a hole you dig.
Aside: in response to the plethora of responses everywhere online about people saying "I compost meat and rarely see rats", I respond: "Rats tend to come out when you're not there. Trust me, they find the meat, unless your bin is under tight lock-down."
Asking a sustainability question about putting meat in a compost pile really is like asking one about the proper tire inflation pressure for a Humvee. True, you can save some gas by keeping tires inflated properly, but if you have a Humvee at all, you've already gone awry. I don't pose this comparison simply as a rhetorical tactic. The direct contribution of agriculture to global warming is roughly equivalent to that of transportation (about 10-15%). If you add in secondary effects from land-use change, it's even worse.
Making the decision to not to eat farmed meat at all is one of those currently uncomfortable sustainability solutions, along with having fewer babies, and not keeping pets. People don't want to consider it, but the impact of the choice is large, and completely within your control, right now.