People mention this quite often to me because my toddler is on oral immunotherapy for peanuts, and there’s a small but important distinction here. It’s extra important when relatives start to think it’s okay to be casually leaving peanut products lying around within the toddler’s reach. (It’s not)
The general consensus among allergists is that early exposure reduces the chances of developing the allergy in the first place, but people on oral immunotherapy are still allergic, they just have a high tolerance and can still have anaphylactic reactions. Some will outgrow the allergy, but for peanuts most don’t and the data doesn’t yet exist for whether peanut oral immunotherapy increases the likelihood of outgrowing the allergy.
There are some early studies out [1] that indicate remission is possible with OIT. (For laypeople, "desensitization" ~= can tolerate some peanut exposure without a reaction, but still needs to carry an epipen and remain on the maintenance dose for life, while "remission" ~= no longer has a peanut allergy). The numbers were 71% desensitization and 21% remission for OIT vs. 2% both for a placebo. It was heavily dependent on age, with 71% of 1-year-olds, 35% of 2-year-olds, and 19% of 3-year-olds achieving remission.
Data will be scant at this time, because the full treatment takes a long time and needs to be adhered to closely. It's 30 weeks of OIT, followed by 2 years of a maintenance dose, followed by a 6-month hiatus to verify whether the maintenance dose can be stopped while still achieving remission, so data necessarily lags the start of any clinical trials by 3+ years.
The general consensus among allergists is that early exposure reduces the chances of developing the allergy in the first place, but people on oral immunotherapy are still allergic, they just have a high tolerance and can still have anaphylactic reactions. Some will outgrow the allergy, but for peanuts most don’t and the data doesn’t yet exist for whether peanut oral immunotherapy increases the likelihood of outgrowing the allergy.