The Demesne, Hearthstone and Manse merits are all intrinsically linked and the rules regarding them are spread between the three. Every Hearthstone came from a Demesne, Though it is considerably rarer for a demesne to do so naturally without a Manse built on it. The stronger the Hearthstone, the stronger the Demesne it came from.
The Hearthstone Merit on it's own grants the character a Hearthstone. These can be selected from the list on p.604 in the Core book or p.132 in Arms of the Chosen (more will certainly be available in upcoming books).
The only restriction is that Greater strength Hearthstones require the 4 dot version of the Merit.
Each hearthstone has magical benefit that it grants it's wielder as long as they posses it. Hearthstones do not require being socketed into an artifact to function unless explicitly stated in their effect. The exception being Hearthstones with the Dependant keyword; these require that they be socketed into an artifact with a second Hearthstone that lacks this keyword.
Hearthstones do provide 'Essence Trickle', as you call it, but this requires being socketed into an artifact. While socketed a Hearthstone provides the benefits of standing within it's Manse. the Manse merit doesn't explicitly state having any effect on those standing in it, but a manse is just a demesne with what amounts to a magic windmill built on it. as such you should interpret it as "if the Hearthstone was manufactured in a manse, gain the benefits of standing in it's Demesne while it is socketed in an artifact". Demesnes provide 2 or 4 motes of essence per hour depending on strength.
Hearthstones with the Linked or Manse-Born tags implicitly form in Manses and stones with the Wild-Born tag implicitly form only in demesnes. Beyond that it's up to the ST and players.