This question pertains specifically to items in the private items/collections: will users be able to search the Omeka default search fields (i.e. Dublin core metadata fields)to find private items/collections if the Solr plugin is installed? I ask because the Solr plugin does not index the private items/collections.
I'm not an expert on this particular plugin, but it seems that your situation would work like this:
SolrSearch doesn't index private items at all, so no users will be able to find private items at all using the Solr search.
However, the default Omeka search should remain exactly as it was before the installation of the plugin. Even if you've altered your theme to make the default search boxes go to Solr, you can get to the "old-style" search at "items/advanced-search".