What means "distribution" for Open Source licenses MIT and GPL? Is the client side code running on user's browser "distribution" for these licenses?
The GPL does not define "distribution" but mentions it as a subclass of "propagation" (which is itself a subclass of "conveyance"):
To “propagate” a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.
To “convey” a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.
In your case, since a copy of client-side code is transmitted to another person, the GPL's requirements about conveyance (notably, supplying access to corresponding source and making the recipient's GPL rights clear) apply to the client-side code you transmit to a user.
The same is true for code under an MIT/X11 license, though in that case your requirement are merely limited to preserving the license text and copyright headers, not sharing the corresponding source.