Translate third party plugin and save translation files in custom theme

First of all, I’d strongly recommend against storing translations for a plug-in in your theme. It has no reason to be there, and you’ll loose the translations when you change/update the theme.

The correct place to store translations is wp-content/languages/plugins/.

This code does that, and targets the domain ‘other_plugin’:

function wpse159536_override_mofile_path( $mofile, $domain ){
    if( 'other_plugin' == $domain ){
         $mofile = WP_LANG_DIR . '/plugins/' . basename( $mofile );
    }
    return $mofile;
}
add_filter( 'load_textdomain_mofile', 'wpse159536_override_mofile_path', 10, 2 );

Of course, an even better solution would be to send the translation file to the developer to include in the next release (if this is appropriate).