This is a question that I've asked many colleagues and although there are only two different answers, I get an array of explanations. Afterall, it is a complex question and thus deserves a complex answer. And like any good debate, the best answer, and usually most annoying is...it depends.
The reason this question is so complicated is because I could point out 5 people training on machines and 5 people using free weights per day and all 10 are unsafe. Again, for the most part, it's not the exercises, it's the application of exercises that matters. However, the issue that I have when I hear that people are "starting out" on machines, is that they assume this is ultimately the safest option. First off, for the most part, anything that puts you in a seated posture already increases the compressive and shearing forces on your spine from that of standing (McGill, 2002). Most machines are poorly designed and already have you seated in a posterior pelvic rotation that we know will consistently contribute to disc bulging and future low back problems.
And that's just the spine. Most upper body machines are still going to allow for a ton of scapular instability, and in fact, if you haven't been trained properly you'll probably default into protraction of the scapulae. When this happens, you are again putting the shoulder at risk of one of many injuries: A-C joint stress, impingement, rotator cuff strains, labral tears, etc.
I hope the "staff" aren't teaching the members how to do a lat pulldown.
I guess the better question is, if both methods are equally safe, what is going to be the best choice for your health, strength and fitness? For 99.9% of the population, this is going to be the use of free weights while standing on the ground...the flat ground! I heard a great quote this weekend from Ori Hofmekler, whose seminar on "Surviving" I attended this past weekend: "All of our ancestors were athletes." In other words, we should be training like athletes: multi-joint, lifting, flexing, extending, twisting, etc. Can you imagine the cavemen sitting on a bench doing a concentration curl or sitting on a leg extension machine working on their vastus medialis?
Cliff Harvey performing a clean grip overhead squat.
So, we as trainer's MUST be able to take a brand new client and begin to teach them the fundamental movement patterns to get them started on a path to a strong, healthy, and fit life. If you cannot squat, deadlift, clean, swing, pull, push, or lunge, then learn from someone who can, go and practice for 20+hours and then teach your clients. But do this NOW...not tomorrow, because in the meantime, your client is teaching herself how to do a prone hamstring curl - possibly the most useless exercise for an otherwise healthy individual.
Training with free weights allows the individual to develop adaptations that will apply and transfer to many other lifts, athletic scenarios, and lifestyle patterns: Resisting extreme spine positions, fighting gravity, neutral and stable scapulae, core integration, rooting of energy into the floor, rotational power, etc. I'm waiting for a machine that can do this. As Stu McGill (2002) says regarding machines, "Training in a proprioceptively starved environment does not challenge the system needed to ensure that no single tissue experiences damaging overload."
Jim Talo performing a single arm kettlebell swing - another great multi-joint lift.
If you are a beginner, seek out a professional - but make sure you do your homework and don't just look up a trainer in some fitness magazine index. Find out their credentials, ask them why they do certain things, and challenge their knowledge. A good strength coach should not have to start you on a machine if you are just starting out!
References
McGill, S (2006). Ultimate Back Fitness & Performance, 3rd edition, Backfitpro Inc.