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But I will never receive that man coming home to the land of his fathers.

Penelope is loosing hope for Odysseus' return.

Odyssey Book 19 line 225-226 Barry B Powell
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Odysseus and Penelope

Odysseus is informing her about 'his' whereabouts:

But give up weeping, and listen to what I have to say. I will tell you truly and I will not conceal it. I have heard that Odysseus will return home soon,

And some later:

He lost his entire crew and his hollow ship on the wine-dark sea when he left the island of Thrinakia.

Thrinakia is the ancient name of Sicily. Odysseus is not mentioning the island of Kalypso for obvious reasons and after that:

But Odysseus, ... ... had gone to Dodona to hear advice of Zeus from the god's high oak, whether being gone so long he should return openly or in secret.

With this Odysseus informs Penelope about his situation. Penelope is a smart woman so, it is to be assumed that Penelope now understands his position and is aware that he might return in secret. And moreover that the beggar in front of her, who has arrived just recently, could be her Odysseus.

She must have recognize some aspects of Odysseus appearance. However Homer does not mention the fact when Penelope recognized Odysseus in the proces of the events following, until the night after the killing of the suitors. And that will be tomorrow on Apollo's Eve.

In fact Penelope is acting like she didn't recognize him untill the last moment, when she plays hard to get.

From this moment on there is a strong interaction between them and they trust each other.