It looks like a bug, to me. Adding some debug message calls, I see that the first n (after p p), correctly calls my-forward-word, but the second n then is interpreted in my-backward-word-map, not in my-forward-word-map.
IOW, the call in my-forward-word of (set-transient-map my-forward-word-map t) seems to have no effect.
(And you get the same behavior if you bind M-n instead of n and M-p instead of p, in the transient maps; IOW, if you try to use only M-n and M-p.)
Unless someone points out something we're missing, you might consider filing a bug report: M-x report-emacs-bug.