We take Beastie down to the Beaches in Toronto with us every day the weather is nice. He started out on a leash...just a little rabbit/ferret leash that was red...and after a couple of excursions, we didn't use the leash any longer. He's happy on my shoulder or on the grass, and he enjoys the ability to run aroud a bit. No, he doesn't go far, and yes, we constantly keep our eye on him. Dogs don't notice him at all - seriously!! They just sniff all around him, sometimes sniff him, and then move on. Beastie also enjoys climbing a couple feet up a tree, and once there, he usually falls asleep. We get lots of attention, and lots of kids pet and hold him. It's funny; 90% of the people that see him say, "oh cool, look at the iguana".
Infact my husband and I just returned from a 20 day road trip. We drove down to Florida and spent a lovely two weeks relaxing in the sun and sightseeing. Beastie, well accustomed to travelling in the car, his carrier (the kind for dogs that looks like a duffle bag - $24 @Walmart), and travelling around, really enjoyed the trip and the car rides. Even crossing the boarder was no issue...just one very simple form from Fish and Wildlife.
So, yes. You can take your beardie out on a leash (or off!), and enjoy all kinds of adventures with him/her. When we're out at restaurants or grocery shopping, we take him in with us in his carrier while he assumes what we call 'incognito mode'...no one has ever asked what was in the bag, and we never told them. And he stays in the bag whenever there's food or eating in public. Enjoy your beardie and know that he/she may be skittish at first. Be patient and always make sure YOU are his safe place.
Annie (& Beastie)