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I would sand the laminated finish. It's not that it has to be a thorough complete sanding, (like if you were refinishing your table or something), but you want to scuff up the entire surface because the glossiness of the laminate will make it more difficult for paint to adhere.
Particularly because it's being applied to a vertical surface.
Melamine paint is probably your best choice, and will adhere the best without having to use a primer. Keep in mind that melamine paint is oil based, and it will smell more when using it, but gives you a hard finish. Also you need solvents like varsol, or linseed oil to clean up your brushes and trays.
You could also use latex/acrylic paint if you prefer (allows you to clean up with soap & water) but I would recommend a primer that is stain-blocking and super adhesive, like 100% acrylic primer. Use glossier sheens for more durability, like Satin, Semi-gloss or High-gloss, and if you want to put on an extra protective layer you can topcoat with polyurethane, either water-based or oil-based.