By default, SQL Server Express does not add a firewall rule, and it has the TCP protocol disabled. Therefore, you need to add these in order to be able to connect.
Open "SQL Server Configuration Manager". In the management console, open the "SQL Server Network Configuration" tab, protocols, TCP/IP, enable. Restart the service for the changes to be applied. Also, you might need to enable and start the "SQL Server Browser" service in "SQL Server Services".
Open Windows Firewall with Advanced Security, Inbound Rules, Add new rule, Custom, Services, Apply to this service: "SQL Server (SQLEXPRESS)", OK, Next... until you can name and finish the rule. Do that again for the Browser Service.
The server name is then the remote computer name instead of ., the rest of the connection string remains the same.
Note that you might run into authentication problems if the accounts are not trusted on the remote computer; for instance if you're not in a domain.