The way the prefixes are typeset is controlled by the internal command
\@character which, by default, sets them in their own line if the
drama* environment is used, and sets them in line if the non-starred version of the environment is used; here's the original definition:
If you want both environments to treat the prefixes in the same way, you can then omit the conditional branching:
\frostspeaks Some say the world will end in fire; \\
Some say in ice. \\
From what I've tasted of desire \\
I hold with those who favor fire. \\
But if it had to perish twice, \\
I think I know enough of hate \\
To know that for destruction ice \\
Is also great \\
And would suffice.
\poespeaks For the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence.