You can paint with acrylics on watercolor paper, and you don't need to gesso or prime it first. You can thin the acrylics to be as fluid as watercolor and therefore transparent. Or you can use them in the consistency as they come out of the tube. They have pigments emulsified in water (waterborne paints), after all, and are not solvent-based.
How well the paper holds up depends really on the weight and quality of the paper as well as how much water is involved in the application and how much coverage the paper receives from the paint. If you want to display the painting, it will need to be framed, under glass.
Consider the Weight of the Paper
You should use a fairly heavy-weight paper—such as 246 to 300 pound—otherwise, you will have problems with your painting buckling as it dries. You may have some buckling and curling while you are painting as well, but the heavier-weight papers will have a better chance of drying flat than student-weight canvas paper or watercolor paper (115 to 140 pound, for example).
You could also look for watercolor paper that is a mix of cotton and synthetic fibers (professional-grade watercolor paper), which may cut down on some buckling.
Or Use a Watercolor Block
If you have a lighter-weight paper (140 pound), you could use a watercolor block to keep the paper from buckling while painting. This also avoids having to stretch the painting. You'll have to wait for the piece to completely dry before removing it from the block, however.
You could stretch the paper as you would for watercolor or tape it down, but if you're using the using the paints undiluted, it would be better to use a stronger (heavier weight) paper anyway.
You could also coat a lighter-weight paper with acrylic medium or a light layer of gesso, which would make it absorb less of the paint. Some buckling and curling could still happen, though.
Decent watercolor paper can be far more expensive than paper made for acrylics, so compare the costs. You may also find that you want to avoid the guesswork of how it will turn out and just use the paper made for the medium, to begin with, if the finished piece will be important. Papers containing wood pulp are best saved for practice.
If you're in the mood to experiment, though, test your techniques on an extra piece of the paper before starting the work, using the dilution of paint and heaviness of application that you expect to use in your final piece. That will give you an idea of whether your paper will buckle and whether it will lay flat after drying.
If It Buckles
Buckling happens when the paper dries at uneven rates in different parts of the painting. To attempt to flatten a buckled painting, lay it image-side down on a clean surface. Mist the back with distilled water (take care that it's not enough to soak through the paper to reach water-soluble paint on the other side) so that it will dry evenly. Cover it with a towel, paper towel, cardboard, piece of glass, or a single sheet of waxed paper (avoid seams), and weight it down with heavy items such as books at least overnight if not for a day. Again, you should likely test the method before trying it on an important piece.