My last employer (private sector) had a division that used 360 evaluations. When asked to provide input to someone else's evaluation I found it helpful when I got either a set of questions to answer or a list of the goals and objectives for that person rather than just a note that I was listed as a superior/colleague/subordinate/customer and please provide feedback. It helped me to frame my responses in the context of the evaluation and give the supervisor the information he or she was seeking.
Providing the questions, goals or objectives when requesting feedback also helped increase the response rate (particularly from those who may not have had a lot of time and/or may have received many such requests).
For the supervisor preparing the 360 evaluation, providing the questions, goals or objectives also helped them get briefer, more targeted and useful feedback for subordinate evaluations. This eased their burden in preparing the evaluations and also gave more significant and specific feedback than just "a great worker" or "a great experience working with this person".
Hope that helps.