To make a report searchable for others, save it in a shared or public folder that is accessible to the intended users. Avoid saving it in the "My Personal Reports" folder if it needs to be visible to others.
PS: View Metadata Search, Salesforce quick search app on Salesforce AppExchange: https://appexchange.salesforce.com/appxListingDetail?listingId=a0N30000009wgF7EAI