Anyone who's able to see an objective can write a progress update on it.
First, pop up the objective. You can do this from your dashboard, the objectives page or on an appraisal report.
Once the objective has popped up, make your progress update at the bottom and click Save:
By default, your updates are Public, but you can click on the Public tag to make them draft and therefore only visible to you.
You can also click on an update to amend it.