Multiple independent Views attachment pagers (load more)
Make sure you have unique pager id's for all your view displays.
You will find the option in Views under the paging section of your display.

EDIT

As noted below, attachments don't have the option to add a unique ID, this is a known limitation. See for example this discussion on Drupal.org.

It is currently not possible to solve this issue in Views only. If you want different paging options, you can not use attachments. A workaround is to use block displays and place them on the same page as the main view. (Do change the pager ID of the block views only if there are still problems).