There are interesting comments on this topic in The New Spencerian Compendium of Penmanship by P. R. Spencer's sons. In Lesson III, on page 12 of the edition I am using, there is a discussion of Ambidextrous Writing. In addition to the benefits of reducing the liability from the loss of the use of the writing hand (including the malady identified as "penman's paralysis"), the claim is made that one actually improves in the use of the dominant hand by the effort to write with the other hand.
The support for this claim seems to be that "ambidextrous writing calls into action alternately both lobes of the brain, equalizing the power of the mind in in the direction and government of both hands. Even the initiatory effort to write with the left hand increases the power of the will in its supremacy over the muscles, as may readily be perceived by the greater ease and freedom with which the right hand is made to execute when it resumes the use of the pen."
I have written with both hands, and enjoy the experience, but unfortunately I don't think I can claim much direct improvement in my writing from having made that effort!