Well, it is going to depend on the amazon and the person. I will just go for the larger Double Yellow Head to begin with on seriousness of bites. They are definitely harder than a conure. Their beaks have much more power and size. I would say if one bites your face you will need plastic surgery (if an actual hard bite not just a warning nip). If it bites your arm I would say stitches will be needed. If it bites your finger then it will depend on the size of your hands, a smaller daintier hand could end up with broken finger besides the immediate flesh tearing. If you have larger sturdier fingers then the flesh would probably be the only problem.
On a scale of 1-10 I would say an amazons bite would rate about a 7-8. It is not as serious as the cockatoo which I would personally rate at a 10 after seeing the damage that can be done. It is not as bad as the larger macaws either due to the size and force of the beak (although I haven't seen many serious bites from a macaw personally). Most usually an African Greys is not as serious as is the conure. They would probably be rated in the 4-5 range with the smaller budgies etc being in the 1-2 catergory. Again this is my opinion due to bites I have seen inflicted and have even received at times. Others may think differently.
Okay now that I have seriously scared some people about being bitten, with the correct handling and watching this need not happen. Of course most bird handlers have been bit at some point and time and more than likely will be bit more times. Usually though bites are more warning than actual true bites. If you learn to watch your bird for signs you will learn to know when it does not want you near or is being moody. Birds will give you signs before they bite even if you don't notice them. They may flash their eyes, fan their feathers, or their tails. The signs are not the same in all birds. Fanning of wings in one bird may mean come play with me and in another may mean come a step closer and I will show you a bite. When you notice the signs and have learned to recognize what they mean you can avoid almost all bites. With a strange bird that seems aggressive then perch training or using the distraction method is more than likely going to be the best bet to avoid any serious bites.
Hope I didn't scare you off of your new addition but I figured it was better to give the worst case and then explain that it hopefully won't happen. Then you won't be surprised if you do end up with an larger aggressive bird at some time.