The answer really depends on whether these users have public or private event data.
For public events that have an associated "organizer profile" page, you can use the organizer_list_events API call without providing any additional authentication credentials.
The event_search method may also work well for finding upcoming public events by organizer.
If you want to access both public and private event data, or events that have ended, then you will need to supply some user authentication credentials. The user_list_events API call is probably your best choice in this case.
The most simple method of authenticating as an Eventbrite user involves providing a user_key in your API request. Each user should be able to find their user_key here: eventbrite.com/userkeyapi/
More info on user authentication is available on the Eventbrite developer site:
developer.eventbrite.com/doc/auth/#user_authentication