Are there any solutions to this logic (?) problem involving modular arithmetic?

There are 10 people in a group, numbered 0–9. Each person draws an integer 0–9 from a hat without replacement. Each person adds their assigned number to the number that they drew from the hat, modulo 10, in order to get a new number 0–9. When they compare their results, they realize that no two people obtained the same number.

Who drew what number from the hat? Are there multiple solutions or no solutions at all?

As an expanded version of this problem, there can be any (natural) number of people in the group, and all of the number ranges are changed to fit the new group size. Is there ever a solution to this problem?