… and I can’t even continue the chat from my phone.
New messages will show on all your devices, but yes, it is intentional that old messages are not available to new devices.
This is because they don’t retain your (encrypted) messages on their servers right? Is this for storage reasons, or more just security philosophy of not being able to access past chats when you login from elsewhere?
This is not entirely correct. Messages are stored on their servers temporarily (last I saw, for up to 30 days), so that even if your device is offline for a while, you still get all your messages.
In theory, you could have messages waiting in your queue for device A, when you add device B, but device B will still not get the messages, even though the encrypted message is still on their servers.
This is because messages are encrypted per device, rather than per user. So if you have a friend who uses a phone and computer, and you also use a phone and computer, the client sending the message encrypts it three times, and sends each encrypted copy to the server. Each client then pulls its copy, and decrypts it. If a device does not exist when the message is encrypted and sent, it is never encrypted for that device, so that new device cannot pull the message down and decrypt it.
For more details: https://signal.org/docs/specifications/sesame/
But if I reply on the phone will it populate the desktop chat and vice versa?
Yes, as long as you set up the desktop client before sending the message.
Messages sent with Signal are encrypted per device, not per user, so if your desktop client doesn’t exist when the message is sent, it is never encrypted and sent for that device.
When you set up a new client, you will only see new messages.
See https://signal.org/docs/specifications/sesame/ for details.
Cool, could I recover a backup to te desktop to have access to the historical ones?
The chat continues on all linked devices from the point in time that they are linked.
Imagine two people having a face-to-face conversation, then a third person walks up and joins in. The third person doesn’t know what was said before they joined the conversation, but all three continue the conversation from that point on.
Linked devices are like the above example, if two of those people were married and tell each other every conversation they’ve had since their wedding.
There is no sharing of messages between linked devices - that would break forward secrecy, which prevents a successful attacker from getting historical messages. See the first bullet of: https://support.signal.org/hc/en-us/articles/360007320551-Linked-Devices
Messages are encrypted per device, not per user (https://signal.org/docs/specifications/sesame/), and forward secrecy is preserved (https://en.m.wikipedia.org/wiki/Forward_secrecy, for the concept in general, and https://signal.org/docs/specifications/doubleratchet/ for Signal’s specific approach).
There is no reason why the message sync that works from phone to phone could not be implemented on the desktop client as well.