This is not exactly how it works. You're right that a big fund executing on a public market will incur (potentially excessive) impact, but the purpose of these private rooms is not to prevent trading against informed parties! Often, the counterparties that a big fund might find on these private rooms will in fact be the same market makers and liquidity providers present on public exchanges.
The difference is that in these private rooms, liquidity providers are often able to understand their customer more. For example, big passive index funds aren't buying and selling due to some adverse knowledge of future price movement. Instead, they are merely following the index. If market makers are able to distinguish between the passive indexers and the smart sophisticated hedge funds, they will then be able to provide to the passive indexers at a better price.
The difference is that in these private rooms, liquidity providers are often able to understand their customer more. For example, big passive index funds aren't buying and selling due to some adverse knowledge of future price movement. Instead, they are merely following the index. If market makers are able to distinguish between the passive indexers and the smart sophisticated hedge funds, they will then be able to provide to the passive indexers at a better price.