Split the project and have two websites
Absolutely to avoid, primary for maintenance. Administrators may switch between the two (same as users with multiple roles) but here you will pay highest price for any new addition/change (and if you pay this then your customers and your users will also pay).
Give the users who have 'roles' in both teams a dropdown in the top right to switch between the two teams
It's less bad than what you may think. Make it clear and outstanding: we are all used to some sort of switch, for example when we have multiple GMail accounts. If you write user name somewhere also don't forget to let group name visible too (or they will not understand where they are):

download bmml source – Wireframes created with Balsamiq Mockups
Note that group name is in a prominent position (top left for most western countries) then hardly it will be ignored.
An example of this?

Don't you think it looks like Stack Exchange navbar? It does and they had a nice design idea, IMO.
any other ideas?
1) Mix your two solutions.
Q. How do you differentiate two separate web sites?
A. Using different URLs.
However you may still have two different URLs that point to same website: www.example.com/group1/newsletters/compose
and www.example.com/group2/newsletters/compose
.
It's easy to do it using URL rewrite (details depend on your specific environment/language). Don't forget you still need a way to switch between the two (not everyone is comfortable writing URL and they may want to quickly switch from one profile to the other).
2) Have different user profiles.
Think about how you login to your Windows domain: domain_name\user_name
(there are even tools to join same computer in multiple domains):

download bmml source
Note that if you use a multi-step approach for login then things will be even more clear:

download bmml source
This second approach is even better because users will not see other groups they do not belong to and who joined just one group will not even see third login screen. Note that first two screens may even be merged together.
To change profile you just need to log out (also do not forget to provide a quick shortcut when applicable):

download bmml source
Hardly users with multiple profiles will be confused because to join a group is part of their login process and they will associate login with group selection. Note that even if you use cookies to keep users logged-in you may still want to prompt them to pick a group for each new access (using a session cookie or Window.sessionStorage
).