OK, you have actually asked for two different dipping sauce recipes as the thai dipping sauce is difference to the vietnamese dipping sauce.
For the thai sauce all you need is
1. fish sauce (100ml)
2. 4 chopped red birds eye chillis
3. lime juice (2 limes)
4. 1 spoon brown sugar
you can adjust the amounts to taste
for vietnamese dipping sauce it is called nuoc cham(slightly sweeter and less hot)
2 tablespoons fresh lime juice
3 tablespoons Vietnamese fish sauce
3 tablespoons water
2 tablespoons sugar
1 garlic clove, finely chopped
1 large red chilli, finely chopped
1 spoon grated carrot(optional)
Stir together all ingredients.