Criminal conspiracies are prosecuted in court daily. To prove a conspiracy, you need to show beyond a reasonable doubt that (1) two or more people agreed to commit a crime, and that (2) one of the people took one overt step in furtherance of the agreement. The overt step taken need not be criminal, and, since crooks don't usually memorialize their agreements in writing, the existence and terms of the agreement can be inferred from their acts. Inferences are not only permissible, they are essential in order to identify and prove a conspiracy took place. However, not just any old inference will do.
Before getting any deeper into the topic of inferences, we need to remember that there are four systems of proof: scientific, mathematical, philosophical (formal logic), and legal. Scientific proof deals in measurable and quantifiable phenomena, while mathematics and philosophy deal with purely theoretical concepts. Only the legal method of proof, which has evolved over several thousand years, tries to figure out what happened between human beings in the past. (The traditional historical method is loosely based on the legal method.) There is always too little information, and what little information there is, is often unreliable. The stakes in legal proof are often quite high, though, sometimes even life and death. Therefore, a longstanding system that can usually make an acceptable high stakes determination, even with insufficient information, is a pretty good system.
Given zealous, competent counsel on both sides, an unbiased jury, and a competent and ethical judge, the legal system usually gives an acceptable result. I am not denying the many failures of the legal system, which happen when there are problems with judge, jury, counsel, or laws. Although appeals take years and can be expensive, they usually make up for flaws at the trial level. While "usually", "acceptable", and "pretty good" are not sufficient for proof for science, math, and logic, they are as good as you're going to get when dealing with human beings and what they did in the past.
Carl Sagan's famous maxim "Extraordinary claims require extraordinary evidence" is the new hurdle in considering conspiracies. What is forgotten is that Sagan was speaking in a scientific context, about UFOs and other paranormal phenomena. Conspiracy is a legal concept and should be analyzed using a legal framework. Inferences in science must meet certain criteria; inferences are allowed to prove a conspiracy in law as long as they are reasonable and have sufficient reliable evidence to support them. Given sufficient reliable evidence of any other reasonable explanation for their behavior, the jury is supposed to disregard the inference that was drawn against the accused conspirators.
Outside a trial, a conspiracy can never technically be proven. But as traditional historians know, the legal method gives us tools to systematically analyze past events and come up with defensible, evidence-based conclusions. Postmodern historians have perfected the ideology-driven historical inquiry, where they take a theory and cherry-pick and misrepresent facts to support their theory. A traditional historian will gather all the evidence relating to an event, weigh each bit of evidence as to its reliability, and from the mass of evidence considered as a whole, try to formulate the best theory that fits the evidence. They then will defend their theory in their peer group through published scholarship and conferences. Through a process of refinement and distillation, a theory most people can live with evolves in that peer group and remains authoritative until the next cultural shift takes place.
Where current thinking about conspiracies gets derailed, besides the skepticism that broad criminal conspiracies actually exist (La Cosa Nostra and Watergate proved such things do exist), is with problems about evidence and inferences. A discussion about how to view evidence outside a courtroom from a legal/historical point of view is a very broad topic, so I'll just go with inferences for now.
Inferences must be reasonable to be credible. A reasonable inference is congruent with human experience and common sense. There is also a requirement of sufficiency: there must be sufficient evidence from which an inference may be reasonably deduced. More credible evidence implies an inference that is more sufficiently supported. An inference may be perfectly reasonable but lack enough supporting evidence, which makes it rebuttable. At some tipping point, there is so much evidence that it becomes unreasonable to deny the inference.
I can illustrate the problem of inference with an example from common human experience. Let's say your S.O. has neglected their appearance for a while; you love them, so you try to let it go. One day they start a diet, get a new haircut, start using skin care products. Their reason is they just got tired of looking like crap. You are happy, so far so good. As they lose weight and start to look like their old selves again, they begin a new activity on Tuesday night. "I've always wanted to learn French." Great, you have, too. No, "this is something I want to do for me". Well, ok, then, we're individuals, after all. A so-called friend, who has always disliked your S.O., casually mentions one day they saw their car outside the Heartbreak Motel. You blow them off, because they've always hated your S.O. and can't be trusted to tell the truth about them. Your S.O. comes home one Tuesday night smelling of an unknown cologne and says they were shopping earlier at the mall and got sprayed accidentally by an overzealous salesclerk. Um, ok. You are starting to get a little concerned, but your S.O. reassures you, and even pays a little attention to you for a few days. But after the next Tuesday meeting, as they disrobe for bed you see their underwear is on inside out. "Oh, I got dressed in the dark and didn't notice." But you didn't get dressed in the dark, we got dressed together and it wasn't dark. "That was yesterday." You think to yourself, no I think it was today, but you let it go because you're not sure. However, even though your so-called friend has every reason to lie about your S.O., you decide to drive by next Tuesday night, and sure enough, your S.O.'s car is parked outside. When confronted, they say French lessons are being held in the conference room. When you ask to go and meet the others taking French lessons, you are told that would really embarrass your S.O. because it would show everyone how little you trust them. The next Tuesday you park outside the Heartbreak Hotel and observe just two people going in, your S.O. and another attractive individual. They also come out together. You confront them in the parking lot, and your S.O. admits that they have been cheating on you all this time.
At some point, probably different for everyone, it became obvious that there was an alternate, reasonable explanation as to what was really going on in this example, even though each individual excuse given was plausible. This tipping point also illustrates another old rhetorical device used by defense attorneys - if you can explain each piece of a damning puzzle with a plausible excuse, then maybe the listener will be too confused or intimidated or unsure to try to put the pieces together anyway and look at the problem as a whole. The answer is that, although each individual occurrence may have a plausible explanation, at some point the totality of the evidence points in a certain direction.
Most conspiracy claims outside the legal system are based on inferences, many of which are either unreasonable, or insufficiently supported. When evaluating claims of conspiracy, the evidence need not be extraordinary, but it should be sufficient enough and reliable enough to credibly support the inferences upon which the claim is based.