Quite a few things to respond to in this thread so far...
First off a shout out to Kids on the Block - i was involved with them in my pre-teens back in the early 80's, nice to hear they're still around and doing what they are!
Early in the thread, an analogy came up to singing - being a puppeteer and a singer, i like that analogy and would like to extend it just a little further..."anyone" (and even then probably with limits) can sing or puppeteer, in the sense that they can pick up a puppet and have fun with it or sing in their shower or enjoy a fave song by belting along with it - they're both creative forms of expression and a form of fun and catharsis. However, the answer of "Can anyone do it" takes on a different tone when you take it away from personal enjoyment to whether one can/should do it for a paying audience. Obviously the answer would be no. Not everyone who has the desire would be able to attain that level.
But if one has the drive and sheer will, that's not to say that they may eventually get to an acceptable level after a LOT of training and hard work...though perhaps years more than what many people may be willing to undertake.
For the person who asked if they can be a puppeteer without voices, of course! Being Muppet fans, it's so easy for us to think as Muppetry as the be-all and end-all of puppetry, but think of all the different forms of puppetry/puppet theatre that has come from other cultures or troupes - there's an awful lot of puppetry performances that don't involve a speaking puppeteer - where the craft is totally focused on the movement. Also think of the more modern forms of puppetry as they pertain to special effects for film - again, lots of opportunity to be involved with the movement of a character where voicing it is not involved. Even if you do wish to think of terms of more traditional Muppetry, if one was to get a job with the Muppets, it's almost a given you would start as a background/"right-hander" where you wouldn't have to worry about voices at that point anyway.
As per the question of how much can be self-taught and learned through trainers...again having experience in the fields of singing and puppetry, i have some definate perspectives on this that parallel each other. A lot can be self-taught and in fact most people who have the desire to be a professional singer or puppeteer started off because they've already on some level "self-taught" themselves since they were a kid. A lot of stuff you will learn will come from that initial intuitive spirit in which you've developed your own style and basics. However, one should never underestimate the importance of a good teacher either - singing and puppetry both are essentially physical acts, and much like any sport or physical activity, a teacher can help with technique and also cure you of some "bad habits" you may have developed by learning something yourself! Looking at the modern protocol of Muppeteers, just about any new Muppeteer will go through extensive training and workshops with veterans (who have in the past/present included Richard Hunt, Kevin Clash, Victor Yerrid). Much can be learned from not only a certified educator but also a working professional with years of experience - someone who knows all about what it's like working with a monitor...having to work in pits on platformed sets...the experiences of working with green-screen, digital puppetry, full-body Muppets, bunraku bits, manipulating small more naturalistic bird/insect puppets meant to provide atmosphere to a scene.
On the singer side of things, i can say i've largely been self-taught througout my childhood/teens and picked up bits and pieces of valuable knowledge from every director i worked with in a choir or show...but when i took my first vocal music course in college and had my first real "training" experience, i can't even begin to place a value on the kind of stuff i learned - not only in terms of technique but in terms of aiding me find my individual style.
It's a personal philosophy of mine that the highest from of art (no matter what type, be it puppetry, singing, visual, writing, etc) is when one finds the perfect balance between the technical ("technique", the actual "how-to"/method) and the emotional (the "heart" "individualism" "expression") To be the best, one needs both parts of the equation - and again, in order to acheive the more technical aspects, you need the training. If it does nothing else, it will give you a second pair of ears/eyes - someone outside of yourself who can see/hear you from an audience perspective and can let you know strengths and weeknesses and how to be better as well as ways of doing things you probably would not have otherwise picked up on your own. But always remember what you're doing is art and if there's not an alement of yourself (ie humanity) in what you're doing, than what you're attempting could easily be duplicated by computers or robots/inhuman technology.
The Muppets fulfill my personal standards/definition of art perfectly - they not only are masters of the craft...all carried down from the true innovator of merging puppetry with the motion picture forms of media, but there's always that certain "spark" that seperates them from other puppet troups that make what they do "Muppet-y" more than any specific trademark. Something that even if its weaker projects, gives them that something "more" than others. You instinctively recognize it as "Muppets" as opposed to puppets. Combine this with the fact that behind the performers are writers, puppet builders, etc etc all devoted to creating something that speaks to people on many levels and you have something really extraordinary which is why we're here.
So there you have it - one of my long convoluted essays on something i get passionate about, but for those who prefer the Cliff Notes version - i refer you to the Fraggle Rock episode "Mokey and the Minstrels" - any Fraggle can (and given the nature of Fraggles SHOULD) sing. But only a particular breed, even among the most gifted, can/should really be a Minstrel.