"How does it work?

xip.io runs a custom DNS server on the public Internet."

Is there a way to provide wildcard DNS without sending internal LAN IP addresses to a closed source "custom DNS server" over the public Internet?

Yes. If you are a djbdns^1 user, and you wanted all subdomains of xip.io to resolve to 10.0.0.1:

   # Assuming _dnscache and _tinydns are the configuration folders and tinydns listens on 127.0.0.1

   cat << eof > _tinydns/root/data
   .xip.io.
   &*.xip.io:127.0.0.1
   =*.xip.io.:10.0.0.1  
   eof

   cd _tinydns/root
   make
   cd -

   echo 127.0.0.1 > _dnscache/root/servers/xip.io

If you are an unbound user, you can put dnscache in front of unbound:

   # Assuming unbound listens on 10.0.0.2
 
   echo > _dnscache/env/FORWARDONLY
   echo 10.0.0.2 > _dnscache/root/servers/@

1 http://cr.yp.to/djbdns.html