Would it be more cost efficient to go to an out-of-state public university or a private university?

It depends on the cost of your out-of-state public university. Some state universities are very expensive to attend if you are an out-of-state student. For instance, Michigan, UCLA and the U. of Virginia will all cost you $55,000-$60,000/year as an out-of-state student. Other state universities will cost much less. Utah State is about $24,000/year for an out-of-state student. Even with good financial aid from a private college or university, you might not do better cost-wise than Utah State. My own experience -- it cost less for my parents to send me to a private liberal arts college (where I got a combined grant/loan financial aid package) than to send my brother to a state university (he was in-state), where he also received a grant.

Out of state is never the answer, unless you live closer to that school as opposed to an in-state choice. Always go public unless a private school offers you a great financial aid package.