Should I get GTX 580 8GB or 1060 6GB

Solution
580 8gb and 1060 6gb are basically almost alike.
One is better on some games, the other is better on other games.
Even if they are different, the difference is not big.
You can simply get the cheaper option.
You question implies that the difference between 8gb and 6gb is a deciding factor.
AMD and nvidia drivers use vram differently so that should , by itself not be a deciding factor.
Each card will perform about the same.
The RX-580 will typically use 75w more power if that makes a difference to you.