I've been using fiberglass screens for years with Ilford paper and have never had a problem of the screen pattern being imprinted on the face of the print.
Are you squeegeeing your prints on the screens themselves? I squeegee my prints on a sheet of glass, and then transfer them to the screens to dry. I gently lay the prints on the screens - not pressing down at all.
I will say that years ago I was told by someone that Agfa Portriga had the reputation that its surface was easily damaged, so I have always dried Portriga prints face up. Other than curling slightly more than prints dried face down, this hasn't presented any problems. So I would take that to mean that if you wanted to dry all your prints face up, you could certainly do so. The darkroom police won't care.
Ultimately, what really matters is what works for you. No one cares if you do things differently that other people.