Reptiles dont have human emotions...when we think of them being "depressed" or "lonely" or something like that we are humanizing them. Reptiles are great, fascinating, interesting animals - but they are not humans, and it does them a dis-service to impress human emotions upon them. Instead you want to try to understand the reptiles for what they are, and cater to their individual needs 
Thats a long way of sasying that no, they won't get depressed
leos dont have emotional attachments to each other and they will be just fine being housed alone.
As Niki says, some people have success housing Leo's together. If you do, you need to make sure you are ready to separate them if needed - if you notice weight loss or fighting etc etc. If you notice any of the Leos losing weight, separate them immediately. I also find that its easier for people who have a fair amount of experience with Leos to house them together - as they are more cued in to the subtle hints that something may be wrong...if this is your first, or one of your first Leos then i really would recomend separating them.
Below i have attached a write up i made awhile back that touches on some of the different issues you need to consider if Leos are housed together. And it also includes a section on things to do if you are housing them together - that will make it more sucessful.
Remember...if you do keep the babies together, and they do well then you will need to separate them in a little while as they get older. This is to make sure that you know what sexes they are before they go back in together.
MULTIPLE LEOS
1. Dominance Aggression/Stress: Many people make the mistake of assuming that aggression is displayed only through physical aggression, but this is incorrect. While its true that these are SOME of the ways that dominance/aggression can be shown there are other, subtler, and probably more dangerous (because they can be harder to notice) ways. When multiple Leo’s are housed together, even females, one of them will be more dominant than the other one, the more dominant one will usually take the best hiding spots and the most food. Getting less food and taking second best spots chronically will lead to one of the Leos becoming stress; chronic stress will lead to failure to grow and thrive and parasite infections. Even if the more dominant Leo doesn't "take" the most food, their presence will often lead to the less dominant one not eating us much. A good analogy is to imagine you are back in school, you are sitting down, about to eat your lunch and the class bully plops down right next to you to eat his lunch...you are probably going to lose your appetite. Right? Now what if that happened every day?
2. Physical Aggression: Not much needs to be said about this, physical aggression is when they actually fight. Chasing, biting, etc. Obviously this leads to drastic problems such as stress, infections and lost limbs. Sometimes Leo’s will appear ok with each other for months or even years and then one day they begin fighting. Any combination of sex can lead to fighting, even two females…especially if you have a female that was incubated at male temps – or a “hot” female, as they are referred to.
3. Positive Sex Identification: This is a problem that frequently gets over looked when people consider putting two Leos together. Leos cannot be sexed with the naked eye until they are close to a year of age. Educated guesses can be made, but there is no sure fire way to tell, without using special equipment. There are a few problems that can arise from this. If you get two baby Leos and house them together and they turn out to be a male and a female, they will figure this out before you do and they may mate, and possibly when they are too young. Mating when they are too young causes serious problems for the female, producing eggs takes a lot of nutrients and energy...energy that young Leos need to be using to grow. Young Leo’s that are bred are more likely to get sick or become egg bound or have other problems related to reproduction. If it turns out they are two males, the Leo’s will be able to tell before you do and you could come home one day to find they have been fighting. It’s possible that one may even kill the other.
4. Disease, spreading and identifying: Another issue when Leo’s are housed together is disease. First and most obvious, if one Leo gets sick, its housemate is more likely to get sick also and then you will have two Leos to treat, not just one. Also, say you come home one day to find a suspicious looking poo (smelly, runny, and nasty) or some vomit, there is no way to tell which Leo is having a problem, unless you can constantly observe them, precious time is wasted trying to figure out which Leo is ill. Or, how do you tell if they are both defecating? If one Leo was to become impacted (or has another issue causing constipation) you may not be able to tell until it’s too late. You will still be seeing fecals, but will be unable to tell which Leo they belong to. Also, one early symptom of disease is lack of appetite, when Leos are housed together it’s harder to tell who is eating how much. This is especially true if you keep a bowl of insects in the tank constantly, or often. You may set out the insects, walk off, come back later and find them all eaten...but who has eaten them? There is no real way to tell, you may not notice one Leo isn't eating until you see significant weight loss.
Its definitely possible to keep Leos together, many people do – and many with great success. The important thing is to identify the risks and take steps to reduce this risk as much as possible. It is also something that is easier to do if you have a certain degree of experience with Leos…with experience comes the ability to pick up on subtle cues in Leopard Gecko behavior, cues which may lead to early recognition of disease. If you do decide to keep multiple Leos together the set up should be as follows:
1. Tank Size: This is very important, a 30 or 40gallon tank should be used when housing multiple Leos, bigger tanks for more geckos, of course. The size is mostly relevant when it comes to having proper hides. You need to have three hides per lizard. For each Leo, there needs to be a warm hide, cool hide and humid hide. Even if all the hides are not frequently used, it’s important they have the choice – no Leo should ever have to choose between being near another Leo and be warm, or be cold.
2. Heat Pad Size: This really is an add-on to the above. You need to have a large enough heat pad to be able to fit three warm hides on it.
3. Feeding: Since it’s harder to monitor food intake with multiple Leos, you should take steps to make this easier. Instead of just throwing in crickets you can either take the Leos out one by one and feed in a separate tank, or feed insects one by one to each Leo.
4. Close monitoring: Always keep an eye out for signs of aggression, stress or weight loss. If any symptoms of disease are noticed in one Leo, that animal should be separated immediately to a quarantine cage. If any weight loss is noted, that animal should be more closely focused on, and potentially removed. Routine weight monitoring with a digital scale will help track any weight loss trends. Any serious aggression should be remedied by separating the lizards. Keep in mind that while some Leos may thrive in a multi-Leo habitat, others will not…so always be prepared to make adjustments if necessary.
5. Spare tanks: Make sure you always have spare tanks available in case a Leo falls ill, or aggression becomes an issue. You want to be able to split them up easily if needed.
6. Ratios: Always combine Leos in sensible ratios, and only house adults in multiple animal enclosures. Only keep one male Leo per any enclosure…one male can be housed with 2 or more females…however you must compensate breeding activity with increased food and supplementation. Multiple females can be housed together. No Leos should be housed together unless the are old enough to be positively sexed.
If you do decide not to separate them...Niki is definaetly right that you will need a bigger tank 
Good luck with the little guys
-----
1.1 Bearded Dragons
2.2 Leopard Geckos
1.0 Uromastyx (Mali)
1.1 Corn snakes
0.1 Mexican Black Kingsnake
1.0 Bairds Ratsnake
1.0 Rosy Boa
1.1 Green Anoles
1.1 House Geckos
0.0.2 Flying Geckos
0.0.1 Red Eye Tree Frog