If I understand you correctly, you'e asking about the "when permanently closed" text, right? If that's the case, only the character that closes the location may use that text, unless it specifically states otherwise. However, after a location is closed, some locations retain usable powers or effects which could apply to any character at that location. For example, the Throne Room has an ability that allows any character at that location to recharge an item in order to draw a card (as long as it's done at the start of the turn). This ability can be used whether the location is open or closed.
Several locations offer a power to the character that closes the location only, but for some locations that power becomes usable by anyone present after the location is closed. Make sure you read the cards carefully!
In the example you stated, only Ezren, as the closer, is allowed to shuffle a spell from his discard deck into his hand. After the location is closed, that ability is no longer available.