Non load bearing wall?

We are knocking down a wall that is not the full width of the room, but juts out into the room just over 3ft. 2 builders have told us that it is non load bearing.
What is the definition of a loadbearing wall?
Our friends a few doors away have removed the same wall in their house and initially did not put any support in place but the ceiling 'bounced' when someone walked upstairs!! They then put in 2 small wooden beams to support the joists.
We are planning to do the same but with just one large beam in the middle.
Will we need planning permission or anything? Our friends did not but that was 2 years ago.

a non load bearing wall is one that does not directly support, lend support to or provide structural integrity to adjoining structural elements of the building.

removing such a wall, resulting in bouncy joists, seems to say that it bears some load.

if the calculated load characteristics of the building are being altered then building control may want a squint.

p.p however will not be necessary.