This feature is available in Enterprise and Ultimate editions.
For the Burndown chart, select Reports from the main menu, and then select Member Dashboard.
The Cumulative Flow report shows the progress at the story level, rather than the task level. Since there may be multiple team members working on a single feature and no viable way to credit just a single person for the group's work, this view is only available at the team level.
Agile, of course, promotes measuring team productivity to encourage collaboration among team members rather than an antagonistic "Don't bother me - I'll do my work and you do yours" attitude. One data point for individual productivity is the amount of Detail Estimate each team member completes in each sprint. Go to the Detail Estimate Trend report to see a summary of the amount of Detail Estimate for each team member on a sprint by sprint basis.