I can only speak for my little corner of the world,but yes,all the privates host competitions open to other private clubs' members--amateur,mid-amateur,senior,junior,and ladies.
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JME,
That's great news. Out of interest which little corner of the world is your 'corner'?
What about the European nations, Aussie, NZ, SA, Canada, Argentina, Dubai, Russia, Chile and all the various other parts of the globe that folks post in from?
Do private clubs your countries hold amateur club 'open' competitions which members of other clubs, or those with proper handicaps can play in? Singles opens, same sex pairs opens, mixed pairs opens, team alliances, junior opens etc?
Oh, just for clarification, most UK club opens are not walk-up's, some are, but most are not......you mostly have to submit an entry form with payment in advance or book online. The website
http://www.golfempire.co.uk/ - is a very good source of tracking down UK open comps although visiting an individual clubs website will also enable you to establish details. Some open comps fill up very quickly, like months ahead, so you have to be prepared to plan in advance.
All the best