No they don't both have to be set. Unless you have some really funky router that I've never seen. I've been using OpenDNS on my macbook for some time now and it doesn't matter where I go, to what network or what wifi hotspot, the DNS still works.
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Just to clarify though, you can do it either way. If you want all your home devices to use it, set it up on the router and set your devices to just auto-use what the router gives them. On something like a laptop that's going to be going to different networks, you may want to explicitly set it so that you can keep using it where ever you go.
No they don't both have to be set. Unless you have some really funky router that I've never seen. I've been using OpenDNS on my macbook for some time now and it doesn't matter where I go, to what network or what wifi hotspot, the DNS still works.
--O