Firstly, the only thing you can / should do with client-side code for the new v1.1 API is: show a timeline using a widget.
Using this widget, if you choose the "search" option, you can construct a search query using your requirements to achieve the desired result.
Head to the create a search widget page, and here enter the following query:
from:user1+OR+from:user2
This'll give you a feed for multiple people.
However, if you want to do this dynamically, you can't use a widget and you will need to be using a server-side language (like PHP) to get the data you require.
If you choose PHP, here's a post a lot of people have found useful, step-by-step with pictures, on how to make authenticated requests to the twitter 1.1 API.
You'll want to be using the endpoint: tweets/search with the previously mentioned query as ?q=from:user1+OR+from:user2
.
This way, you can inject the username into the query dynamically within your PHP code before shooting off the request - how you perform this architecturally is entirely up to you, but the above post and library are a great start.