The position of 'really' changes the meaning of the sentence 1. He's just a puppet in the company and "doesn't really" have much authority to make decisions. In this sentence you are using 'really' to modify the word 'have'. By doing this, you are emphasising the fact that although he appears to have authority, in reality he has very little. 2. He's just a puppet in the company and "really doesn't" have much authority to make decisions. If you emphasise the word 'doesn't' by placing 'really' before it you are stressing the negative. This means that he has virtually no authority. Both are possible. However, only version #1 explains the meaning of 'just a puppet'. So if you have to choose one of them, you should choose the first. The second option is not wrong in this context. But the first option, which has the meaning of 'his apparent authority is a sham' would be incorrect in other contexts. Placing the modifier before 'have' creates a meaning specific to this context. I hope that makes sense!

Both are correct. You could actually remove 'really' from the sentence and it would sound a little more formal if that is what you want as well.