I am just rekindling my interest in Ancient Rome, and reading Rome:Its people, life and customs by Ugo Enrico Paoli. He goes into considerable detail about tables, but only dining tables. There is a whole chapter on writing implements, tablets, parchment, papyrus, stylii, pens,ink etc, but nothing about writing tables or desks.
It's perfectly possible to write on a tablet or slate either sitting or standing, but not on parchment or papyrus, needing a flat surface and somewhere for pens, ink etc. And - the Romans wrote a lot - letters, books, laws, poetry, treatises, orders etc. They even had teams of literate slaves writing popular books to dictation. And, not just writing - accounting - one needed a surface of some kind for an abacus or scales.
My admittedly small Latin dictionary is no help, it translates "desk" as "scrinium" which then translates as "book box"!
So - did the Romans have desks/writing tables? They were a practical people, they must have solved the problem somehow!