If you rewrite the words that a person said, then you must take the quotes off -- but you still attribute it to the source. This is called a paraphrase.
Let's say you interviewed a woman named Judy Chang about a cement factory. Here is what she said to you (in part):
"And the water will be -- They'll dump stuff in the water."
You can paraphrase that this way:
She expressed concern that the factory might pollute the water.
A paraphrase does NOT have quotation marks around it, because it is not what the person actually said.
You are obligated to be faithful to the intentions of the person you talked to. Make sure that any paraphrase you write accurately conveys what the person meant.
Judy Chang also told you this:
"I think the cement plant is a bad thing, well, not a bad thing exactly."
It would be inaccurate to report only that she said it was bad. She clearly did not mean to say that the cement plant is bad, period.
If it's not what she meant, then it isn't true. Journalism must be true, factual. It's not always easy to hold to the highest standards of accuracy, but that is exactly what a journalist is supposed to do.