I live in a state where all high school students have to take the ACT. Are my scores from the state-mandated test ok to submit to colleges? Or do I need to take the ACT again independently and submit those scores instead?
There is no difference between a state-mandated administration of the ACT and an ACT that you take independently (besides the fact that you won't have to pay for the first kind). There is usually no need to take the test again if you're satisfied with your score from the ACT administered by your school.
The only exception is if your state ACT doesn't include the Writing section and your colleges of choice ask that you submit Writing scores. States that administer the ACT without the Writing section include:
Colorado
Kentucky
Louisiana
Mississippi
Tennessee
Utah
Two other states where you may take the ACT without Writing in school are Arkansas (it's determined by district whether students have to take the ACT or not, but it's always without Writing) and North Dakota (where students can choose to take either the ACT without Writing or ACT WorkKeys.