If you have a legacy account on Poloniex (created a year or more ago), then you have 2000$ daily limit for withdrawal, otherwise you must go through verification.
If you want to withdraw from bittrex you must do verification for all (old and new) accounts.
They have basic verification, where you just enter your name and address. It's meant for people from US, but almost never works.
Then is an enhanced verification (for all other countries) where you must send them images of your IDs and selfies, which must match entered data in basic verification. If you don't want to send them this (due to fear of identity theft and similar), then your assets are stuck there and you don't have any means to withdraw them (they've basically stolen your money).
From my current perspective, Poloniex did far better migration to KYC. Bittrex messed a lot of things up with KYC implentation, on top with veeeeery slow support.