How do I grant permissions for translating block content?
Block level permissions for administration is not provided in core. There is a module Block Access which provides block level permissions for admin interface.


  This module adds a set of global permissions for creating, viewing, moving, enabling, disabling and configuring blocks as well as permissions at the individual block level. These permissions only affect the admin interface, front-end view access is handled in core.


UPDATE:

Blocks are not content, so the permission to translate content won't help here. To be able to translate blocks you need to download Internationalization and Variable modules and enable the module Block languages [i18n_block] which ships with Internationalization module.