you're right. I just checked right now, it has nothing to do with the dampers striking on the keys. But it is something of the pedal mechanism. If I just press the pedal, without even touching the keys, I hear the strings vibrating (and the noise, of course).
Because you say "and
the noise", it looks like you are describing two different problems. You have the pedal mechanism making one kind of noise, and you also
have "the strings vibrating" when the dampers lift off them. This second type of noise is, I think, what Monica means by "swooshing sound", and it is usually much more noticeable when the dampers lift off the strings than when they land on them. It is presumably caused by very slight friction between the surface fibres of the felt dampers and the strings, and it will probably never go away, it will just gradually become less noticeable as the fibres settle down, so it's a problem which is much worse on new pianos than on ones which have been played for a few years.
It may be that the mechanism noise also then echoes around the strings, so when you get the first problem fixed, perhaps the second one will be less bad.