I was curious what people mean when they say you need formal training in a field?
It means that this person has received explicit, recognized training, following the norms of the field. For instance, for a physicist this may mean having a university degree in Physics. For a medical practitioner this may mean having gone through med school. For a Linux sysadmin, it may mean having some recognized certifications.
The term is usually meant in contrast to learning through self-study or experience. For instance, we would say that a software developer has formal training if he has a degree in Computer Science, but not if he learned his trade solely by working through many online tutorials or through practical experience working in open source projects.
Formal training would be learning something by completing a degree, certification program, coursework or other formal program.
Informal training would be learning something outside of an official program. You might learn something on the job, teach yourself, or have a friend teach you.
Depending upon the specific context, there is a grey area as well. For example, a graduate program my only consider formal training to be university coursework. Or, learning on the job may give you a company-level certification. Depending upon the company, their reputation, and the program's rigor this may be informal or formal training.
For example, McDonald's has Hamburger University that provides training to company employees. Insides McDonald's Cooperation, this is would be formal training. Outside perspectives would vary depending upon who you talk. An academic program likely would not consider this to be formal training.
Within any one specific circle, it usually boils down to accountability for whoever might be employing that person or using their services. If your pharmacist says "this is X, take 2 a day for a week", you'd expect there to be a big government body to have signed off to say "She seemed to know her stuff, you can trust her" and its really this that you trust, not the stranger at the counter. How 'formal' a program is, is usually synonymous with how good "She must be good she did ". It's worth noting that its very rare that the training itself is the formal bit. It's usually the test at the end, whatever that maybe. Nobody's allowed to become a doctor because they went to medical school. They have to pass the course.
What about in general? There it means surprisingly little! There are very few fields that universally acknowledge an accrediting body. Let alone between fields. Both within and between fields the standards vary wildly. @RichardErickson's example of Hamburger university is a great example.
This is inescapable and perhaps desirable, as it should depend on the gravity and nature of the particular circumstances. There's not much need to prove you know what your doing to do crypt-analysis if you get somewhere you already have the proof, and that was the aim. On the other hand, if you were implementing or auditing an existing system, it's important for whoever hired you to be confident you met certain standards. Equally "formality" of training for bar tending is less important than for pharmacy. Getting the wrong beer is less problematic thatn getting the wrong drug, so the chain of trust is less important.