Prismacolors and Copics are both alcohol-based so a person *should* be able to color in an image and have no streaks/spots.
However it can depend on the cardstock being used, speed of coloring, etc etc.
Kitty's advice about coloring faster is a good one. Or at least get all your base color down and your darker colors (if you're blending and whatnot) at your own pace THEN go over the image quickly using a tight circular motion keeping in mind that you want to keep a "wet edge" and you should be able to eliminate the streaks.
And honestly depending on what you're coloring a bit of streaking or a non-perfectly blended edge prolly won't ruin an image. Sometimes the things that we notice as the crafter and are irked by are likely if ever noticed by the recipient