Every vet has to start somewhere. Heck, every doctor, hair stylist, lawyer, or other professional always begins their career with limited experience. At some point they are going to be faced with a situation, procedure, or case that they have never encountered before and have to handle. That is always nerve-wracking for the professional, but can also induce fear if the client learns that this is their first time.
One of my associates is a new graduate and recently encountered this situation when a client freaked out a little after learning that the doctor had never done a certain procedure before. I certainly understand the caution and apprehension from the client, as nobody wants their pet to be the first time a doctor performs a surgery, does a test, etc. I'll be honest in saying that even I would prefer a doctor who was experienced with a given procedure rather than one doing it for the first time. However, somebody's pet must be the first one for the vet, otherwise they would never gain the experience.
In veterinary school we do get practice at many procedures and surgeries. Some schools have programs where a student performs many spays and neuters before graduating. With more complicated procedures it often depends on what cases come into the clinics while you are on that particular rotation. For example, when I was on my two-week orthopedic surgery rotation I ended up helping on two total hip replacements but no ACL repairs, even though the latter surgery is much more common. But even with this experience there is no way we can get hands-on practice on every procedure we might perform in practice.
I remember the first time I ever performed a "cherry eye" surgery (prolapsed tear gland). I had seen them done a couple of times and when I looked in the text book it seemed like a fairly simple procedure. Because I had basic surgical and tissue handling skills I felt confident in my ability to be successful, and went ahead with the surgery. The case turned out great and I've done it many times since then. This wasn't my "first" in practice, either. There are many surgeries and other services I've done for the first time after looking it up and studying prior to being with the patient. All of them turned out well, in part because I try to be honest about the limits of my skills.
I know this may sound scary to many clients, and we veterinarians don't typically say "this is my first time doing X" because we're aware of that fear. But every vet has a pet that is their first spay, their first hernia repair, their first tumor removal, and so on. Because of the nature of how cases come in we may not have the opportunity to watch or assist in a procedure before we're called on to perform it. If we're in a good practice we'll have a mentor doctor that can watch us do it or be available to jump in and help if needed. For example, my new associate recently was faced with a case where the dog needed a tooth extracted. She had never done this before so I did it for her with her watching. The next time she'll do it with me watching, and after that she'll do it without observation. This is often as much "training" as we get in many procedures.
So why aren't we trained more? It's simply a matter of time and cases. If we didn't graduate until we had done 90% of the things we'll do in practice it would likely double the education process. We do as much learning on the job and under the tutelage of an experienced vet as we do in school. This is true of all professions, as there is only so much you can learn in school before you have to go out and actually do things.
Why is this safe? Isn't there more risk to a patient if the doctor has never done the surgery before? To some degree, yes. The first several times a procedure is performed it takes longer and there is less skill. You have to do something multiple times before you get really good at it. When I was a new graduate a spay took me at least 45-60 minutes and I often had bleeding due to sutures that had slipped (which I always identified and fixed before closing the patient). Now I can do the same thing in 15-20 minutes and have "bleeders" once or twice a year. But if I hadn't started doing those "first" spays, I wouldn't be as good of a surgeon as I am now.
Another reason this is safe is because of the extensive training we do receive. Much of what we learn is general principles rather than specifics. In surgery we spend as much time learning different suture materials and patterns and how to handle different tissues as we do with a single surgery. Once we understand the general ideas and tissue handling we can extrapolate that knowledge to other areas. If I know how to remove a skin tumor on the chest, I can also do it on the ear. And if I learned how to do surgery on intestines I can use similar methods on the eyelid. Certainly there are procedures that should be learned first-hand by a mentor (such as orthopedics), but that is not true of ever case. Much of our veterinary education is based on learning the broader principles that we can apply in many circumstances rather than a single instance.
To new vets.....You know more than you realize and don't have to have witnessed every single thing before doing it yourself. Yes, it's scary, and be aware of your limitations, but don't be afraid to do something just because someone isn't there to hold your hand. That being said, you'll gain skills and experience quicker with a mentor, so find a practice that is willing to take you under the wing of an experienced vet.
To pet owners....Don't get too worried if your vet hasn't done a given procedure before, especially if they seem very confident about it. By all means ask lots of questions and get a feel for why that vet is comfortable making your pet their "first". But keep in mind that every single procedure we've ever done had a "first", and the huge majority of those turned out great.