Simply put, you need different paged variables for each paged element on the same page. Otherwise, like it happens for you, they will use the same and logically be on the same page.
The codex entry for paginate_links() gives a basic example for the usage of the paged variable. Now for an explanation on how to set up a additional variable for a secondary page-able area take a look at my answer to »
How to paginate attachments in a secondary query as gallery?
«, which shows all important steps you will need to do and should get you started – at least. Last but not least, there are surely more Q&A’s on here that will help you, you just have to search them.