should only skeptical witnesses be checked out?

If something highly odd happens then it is rational to ask why it happens if it really has happened.  A sceptic is not a scoffer but a checker. It just happens that the checking out shows nothing really supernatural happened.
 
The more out of the ordinary a claim is, the better the evidence you need for it.  To declare Bigfoot real, witnesses and footprints are no good. They are only good for showing the subject needs debate and discussion. You need a body live or dead. You need that for the risen Jesus too and even more so for nobody is calling Bigfoot supernatural or a miracle of God. Extraordinary claims need extraordinary evidence - meaning hard evidence.
 
For example, suppose people supposedly saw the Virgin Mary in an apparition. The people must be kept apart and questioned as soon as possible. Even a few days could be too long. Tests done to eliminate the possibility of drugs or mental illness would be an essential. Their background would need to be checked in terms of honesty and truthfulness. This is very far from the notion: extraordinary claims require extraordinary evidence. It is merely good practice. And if God is behind an apparition he will ensure that these steps will be taken. He governs all things and influences all things. But apparitions never meet it.
 
One thing is for sure, if you need good practice to show that a male dog has a womb you need better practice in relation to a miracle claim for that is far more out of the ordinary.
 
We are no doubt obligated to try and find truth. We must accept it when we do.
 
Christianity says the biggest obligation in all that is to believe in divine truth for God comes first. He is God of honesty, love and truth. Love cannot exist if there is no concern for truth. So the truth of God and about God matters most. It is more important to know/believe that than to know/believe that antibiotics save lives. Thus if extraordinary claims need extraordinary evidence, then extraordinary claims about God and what he has done need the most extraordinary evidence possible. It is disrespectful to God, if he exists, to deny that. It is being an enabler to ideas and faiths that are simply man-made. Man's word becomes the pretended word of God.
 
The best evidence comes from honest witnesses to miracles who prefer non-miraculous explanations. If they are wrong that it is not a miracle that does not matter. Their evidence will still show that it is. The evidence of the believer violates the rule that the evidence has to be exceptionally good. People tend to interpret things according to their preconceived ideas.
 
The best educated person seems to be the best witness of a miracle. The person must know psychology, science and magic tricks to make an educated decision that she has really encountered the supernatural. It never happens though that such a person sees or experiences a miracle. And you see extremely intelligent books that contain the most outrageous rubbish. For example, think of the top philosophers in Catholicism who will have superstitions they endorse such as Mary being able to hold God's wrathful hand back and communion wafers being actually a living breathing man. The smart qualified person who experiences a miracle is not necessarily a convincing witness but would be a better one than the fools the Church honours as people who have witnessed miracles and whose witness is justification for belief in miracles.
 
Miracles that few want to believe in are soon forgotten. If people want to believe, that is when the miracle gets attention. And the Church and the investigators may come along to check out the miracles if the attention is great enough and publicised enough. The more people want to believe or the more people who want to believe, the greater the chance of self-deception. Even the most honest of people sometimes want to fool themselves and be fooled. You can never even partly tell if a person is telling the truth about experiencing a miracle. All you can do is assume. You can't just assume a miracle report is true. You need evidence.
 
Rational people would not examine only popular miracles.
 
Religion needs to reject or at least ignore the testimony of miracle believers who do not understand the following cardinal religious doctrine, "God comes first. He forbids all lies. Thus if our telling the truth, hurts people that is sad but God comes first and it is a necessary evil." The Roman Church has never checked out its miracle witnesses knowledge of the doctrine. Knowing the doctrine would not prove that the person is telling the truth. But it is better to heed the person who takes the doctrine seriously than a person who doesn't even know about it or understand it. There is a bias that prefers heeding people to God. For example, religion claims to be revealed by God. But religion chooses to believe what prophets and witnesses to miracles tell it about God. That is not the same as hearing God himself. It's hearsay.

Even if no miracle claim should be taken very seriously, if you are going to take miracles seriously then look at ones that survived sceptical inspection and this inspection needs to be happening when the miracle is ongoing not years later.  Any miracle claim should be dismissed that is just based on what believers say.  Nothing in the New Testament indicates that any sceptic knew of the resurrection claim while the appearances were allegedly happening. Thomas is not a sceptic for he did not weigh the evidence for and against but just laughed at the resurrection reports.  No unbeliever said at the time that Jesus seemed to have come back from the dead and he didn't know how.
 
The sceptical witnesses should be checked in preference to the pro-miracle witnesses.