They didn't want Nick Locarno in "The First Duty" because using him as a main character on VOY would mean Paramount would have to pay "The First Duty" writers every time they used the Locarno character. They didn't want to do that so they created Tom Paris for Robert Duncan to play.
Wouldn't any character they wrote for the series be property of Paramount at the time ? Isn't that standard practice, that the stuff you write for the series isn't yours, like programs you code for a company aren't yours ?
Characters created for the series did become property of Paramount, yes. Therefore, they could use them again without getting the writer's permission. However, the standard Writer's Guild contract for writers who are not regular staff writers specifies that royalties are to be paid to the writer any time one of their characters is used in a subsequent episode. While Ronald D. Moore was a staff writer at the time of "The First Duty," Naren Shankar was not.
So, yes, Paramount could have used the characters without getting the original writer's permission, but they would have had to pay him.
That being said, I believe more has been made out of that aspect than was really the case. I think it was more a case of Locarno not being the exact character they wanted on Voyager, with the royalty issue being a contributing factor.