Short answer: No. Longer answer: There is no evidence for this. This is despite the scientific paper and BBC article that claiming that obesity can indeed spread from friend to friend. The original work was in 2007 by Christakis and Fowler and appeared in the New England Journal of Medicine (NEJM). NEJM is a very prestigious medical journal.
Christakis and Fowler are trying to do something very hard. What they are trying to do is assess if you can somehow ‘catch’ obesity from a friend, i.e., if while all other things remain equal, if a friend piles on the pounds, this necessarily makes you significantly more likely to also become fat.
A basic problem with proving (or indeed disproving) this hypothesis is that indeed it is quite likely that if you are fat so are your friends. But that is correlation not causation, and correlation and causation are very different things.
The key point is that we resemble our friends. You should not be surprised to know that I have a lot of friends who are scientists and that a lot of my friends have PhDs. This does not mean PhD’s are contagious. Sadly for my PhD students to get their PhDs they have to write their theses and pass their vivas. Simply being in the same room me as when I sneeze will not be enough.
The fact that we resemble our friends is a problem for any study that aims to show something actually spreads from one person, i.e., here that obesity in person A actually makes it more likely than his friend B is also obese, as opposed to A and B just bonding over a mutual love of Big Macs, and supersize fries.
Distinguishing between fat people naturally coming together (correlation) and fatness in one person causing fatness in another (causation), is very very hard. Basically what you would need to do is somehow compare two groups of people that identical in every way except that in one group the people have an overweight close friend and in the other they do not. These two groups would have be identical in all other ways, same education, same balance of men and women, etc etc. Because all these things can influence the probability of becoming fat.
Christakis and Fowler don’t try to do this, they don’t have the data. Instead it looks like they try to blind the reader with fancy stats, which are critiqued here by Lyons. Lyons also says intelligent things about many medics (and others) having a poor grasp of stats. Which is a problem.
PS A note on error bars, if you read the BBC article you will see “risk [of obesity] was increased by 57%”. In the original article, error bars are given. Big error bars. They authors say that there is a probability of 0.95 that the true figure is between 6% and 123%. It is a bit worrying that journalists are neglecting error bars of that size.