Stress is part of every dragon's life, anything that makes it nervous etc, will cause stress. How long that stress last when it occurs and how often stress comes up, can cause health issues.
For the most part, a dragon isn't stressed if all its base needs are satisfied, such as proper basking temps, UVB light, etc.
Dragons under a year old can stress easy. They instinctively see anything bigger than them as a potential predator. Thus any cat, dog, person, bird, large moving shadows (like a spinning ceiling fan) can stress a dragon easily. However, dragons will adjust to sources they are not sure of in time. Thus regular handling and interaction will reduce causes of stress and accustom your dragon to you being something other than a potential predator.
The real danger with stress is prolonged stress can cause secondary issues, which lead to health problems. It doesn't mean having the neighbors dog over for a weekend while you petsit for them and it spends all its time staring at your dragon once in awhile before wondering off is going to cause your dragon to drop dead. Of course, if the dog tries to get at your dragon, barking and jumping at the cage till it either breaks through a screen or knocks it off the table..stands a high risk of killing your dragon (due to physical trauma).
Taming a dragon does take persistence and patience, as already mentioned. Dragons under a year old are in some ways very easy and very difficult. They often act very calm and tame for months then go into a 'don't touch me' phase full of hissing, bluffing, false attempts to bite and even real attempts to bite. Not to mention severe twisting and clawing when finally in hand. This tends to freak people out as they wonder why little 'dragon-poo' was just a darling a couple weeks go and suddenly is this little terror. Its called puberty. Not all dragons are this extreme though but brought up the extreme case to show the better contrast of behaviors.
Bearded dragons are not dogs, who can quickly warm up to people, even strangers within a very short period (anywhere from 1 min to several weeks). Bearded dragons do not associate humans as 'friends' like dogs as bearded dragons have not been pets for 1000's of years, as dogs have (and cats as well) and bearded dragons are not pack animals like dogs. It is a reptile.
Thus it takes much longer to get dragons used to being handled or being around humans. Hand feeding certainly helps, talking gently and making sure the experience with you is pleasant and rewarding (using baths is good, unless your dragon hates it..in which case I don't recommend trying to make it the 'happy time' for your dragon). Good news though, is most dragons can be conditioned to not fear humans and enjoy your company. However some dragons won't tame down completely but these are relatively rare and most often come up with older dragons that were not held by humans at all or very much (or worse, abused by previous owners).
Last note. Keep other household pets away from your dragon. Though it is possible dogs and cats will just ignore your dragon, it only takes a second for your cat or dog to decide the bearded dragon is no longer something to ignore and take a bite out of it. Cat bites or scratches can cause death due to secondary infections easily and often the wounds are so small they go unnoticed till too late. Big dogs could swallow a dragon whole or in a few quick chomps. At the very least, don't leave a cat or dog alone with your dragon, in or outside its cage (too many times have cats gone through screens and fell inside cages, maybe not actually trying to get your dragon, it could simply have jumped there to enjoy the warmth and its weight ripped through the screen). Knocked over heat lamps can also be a fire hazzard so safer to keep cats and dogs outside the room bearded dragons are kept in when you are not around. (even at night I suggest keeping dogs and cats locked out of the room reptiles are kept in while you are sleeping).
I cringe when I see photos of a cat or dog with a lizard or snake lying on them or sleeping on them...I just hope the owner whisked the reptile to safety after taking the 'cute' picture. Too many times I Read on this forum and others of the cat or dog that broke into a reptiles cage and now the animal is either injured (from bites, or just having the cage fall over) or worse, missing, fate unknown. (ie eaten or just escaped)
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PHLdyPayne