Because the writing requirement explicitly mandates that you receive feedback on a draft of your paper, you have a built-in advantage that those writing an independent Note for journal submission do not: formal review by a faculty member—and likely one with an expertise in your topic area. That opportunity is invaluable, and you should make every effort to generate a draft that is conducive to meaningful substantive feedback.
If you are considering requesting review and feedback beyond what the faculty member provides, it is helpful to think about feedback in two categories: 1) general and 2) substantive.
Though general feedback (e.g., grammar, spelling, structure, clarity) is key for any writing project, substantive review is especially important in the scholarship context. Substantive reviewers should be able to vet the soundness of specific arguments, identify any gaps in the proposed reform, refer you to scholarship that you were not aware of that is directly relevant, etc. Solicitation of substantive feedback will greatly narrow your audience for review, but it is the more important of the two forms of feedback.
Outside of Written and Oral Advocacy, it is perfectly acceptable to reach out to those in your network with a solicitation for general feedback on your paper. Provided that the work remains your own (i.e., recommendations to improve clarity or word choice do not include prescribed text amendments), you should feel free to request a review from a peer or mentor (if appropriate to the relationship).
Who to ask:
When soliciting feedback be sure to include the scope of the feedback requested: if you are not interested in copyedits, be sure to say something to the effect of "If you have the bandwidth, it would be incredibly helpful if you could provide a cursory review..."
If you are requesting specific feedback about grammar, spelling, and clarity, be sure to give the reviewer sufficient time to provide the feedback--nobody appreciates being asked a favor that also comes as a last-minute request.
Though substantive review is essential to legal scholarship, it can be hard to access as a law student. That is why it is important to take advantage of the built-in review process that the writing requirement mandates. Communicating your request and expectations clearly, often called "managing up," is the best way to access meaningful feedback that helps you to refine your paper:
If you wish to seek additional feedback on substantive matters (e.g., if you are writing a California Constitution paper and your topic is focused on water law, you might want the paper vetted by a water law professor), think about identifying 1) someone who clearly has expertise in the topic area (it is probably not enough that they teach international arbitration, if their practice specialty is in employment and you are interested in the arbitration of environmental disputes) and 2) the nature of your relationship. Though many law journals regularly solicit feedback from professors as a review safeguard, it is less common for students to do so independently: tread lightly, politely, and keep expectations in line--particularly if you do not enjoy an existing level of familiarity.