As a conservative Reformed church guy, who has been a member of both Scottish and Continental variants of this corner of Christianity, I must say the table is ALWAYS fenced. But thusly (or something close): "If you put your faith in Jesus as your Savior and are a member of a church that preaches the Gospel, you are welcome to the table." I think that is pretty much wide open for all Bible believers.
I remember attended a Reformed Baptist church one Christmas Eve and they had a communion service after the main service and the deacons examined the participants. We stuck around note being sure if they would admit paedobaptists to the table. He asked me why we should be able to partake and I told him, answering for my entire family, that we put our trust in Jesus alone for our Salvation. He inquired about our church membership and I told him we were members of "Acme" Presbyterian, PCA. He shook my hand and welcomed us to the table. I understood why they fenced the table, out of concern for those who may be taking it wrongly. But the question was quite simple and they clearly weren't excluding other Christians.
That's a little peculiar...Bible doesn't say to examine others...It says to examine yourself...