Are we talking African or European Swallows? And are they laden or unladen?
But on a more serious note, you would need to specify a species first, calculate the average liftable weight of that species and then devide a persons weight by that number. No doubt it would be a hopelessly inacurate figure, but it would give a clue at least.
Assuming a somewhat more conservative figure of 10 lbs per bird and average American human weight of about 190 lbs (via http://en.wikipedia.org/wiki/Body_weight), it would take about 19 bald eagles to carry a person in flight.
However, I doubt you'd be able to use the "James and the Giant Peach" approach of simply tethering each bird with a string/rope. Chances are the birds would be too close together to actually fly. So, you'd have to build some kind of framework to keep them separated, which would add significantly to the weight they're lifting.
Ultimately, it could probably be done but you'd have PETA and Greenpeace and others of their ilk banging down your door if your little experiment became public.
Begin with a human weighing 150 pounds.According to this encyclopedia (http://answers.encyclopedia.com/question/much-weight-can-eagle-carry-85460.html), golden eagles can lift the most weight, and they max out at around 8 pounds. To be a safe flight, I'll say 6.5 pounds is average. Then, there would have to be some kind of structure built to harness the birds without obstructing their wing-span, and without tugging on their ankles (bird ankles?). Lightweight enough to offset excessive additional weight requiring too many extra birds. Titanium? I think this is a good option. Nope. Aluminum is lighter. Well, really, aluminum is lighter than titanium, but titanium is stronger, which allows for making it much thinner. I'll use titanium. Thin and light. (see question 4a here: http://www.lashersport.com/faqs.htmlhttp://www.lashersport.com/faqs.html)I will build a harness in some kind of flexible link type of design, that has small, upraised, well-positioned posts with neoprene covering them, that are just large enough for a bird's talon to encircle properly. Each bird gets two posts to latch onto, with their talons. They can fly down in a great flock and each take their positions and then, lift off. Meanwhile, our human will be attached below with a somewhat flexible bungy-ish cord, so she/he can free float nicely through the air. My titanium harness, with neoprene and bungy-ish cords would weigh an additional 10 pounds, approximately. Add 150 + 10 = 160At 6.5 pounds per bird, that is 24.61534615384625 golden eagles. I will add an additional 2.385 birds to be certain of success, for a grand total of:27 GOLDEN EAGLES to lift a human in to the air. Ha!.
More like insteps actually, anatomically speaking.
As for your light framework, how about carbon fibre composites?
Hmmm...Or even bamboo... We get too obsessed with modern materials. Bamboo and string technology has reached high standards of simple but sophisticated design and application. I understand that when they were making the film "Those Magnificent Men in their Flying Machines" they had serious problems trying to replicate the machines of the pre-1914 generation. Constructing good bamboo-and-string structures required some forgotten techniques. But workmen competent in the field can produce results of a quality that compels respect in perceptive critics even today.