This works:

But here I don't get the chance to write the passwords.

How can I fix this (i.e. make the first sudo block properly until the password is supplied)?

Using sudo in the local machine will give you ZERO privileges in another machine, per se. The only situation when something like that would happen is if root itself has a ssh login enabled in the remote machine (and this is an extremely dangerous thing, unless it's somewhat mitigated by using keys and PermitRootLogin=without-password in /etc/ssh/sshd_config).

Furthermore, assuming you stopped using sudo in the local machine and wanted to pipe the password thru ssh to a remote sudo, it will also fail because sudo only read its password from a tty, which in the case of a normal ssh connection isn't allocated:

me@localhost:$ echo hello | ssh user@remotehost -t sudo cat
user@remotehost's password: ******
sudo: no tty present and no askpass program specified

You *could* force a tty allocation by ssh using the -t switch, and then type the password when the remote sudo asks:
me@localhost:$ ssh user@remotehost -t sudo echo hello
user@remotehost's password: ******
[sudo] password for user: ******
hello

But this won't work when you want to pipe things to ssh:
me@localhost:$ echo hello | ssh user@remotehost -t sudo cat
Pseudo-terminal will not be allocated because stdin is not a terminal.
user@remotehost's password: ******
sudo: no tty present and no askpass program specified

I'm aware that that sudo in the local machine will give me nothing on the remote machine.
The example I gave serves its purpose, but it's not exactly what I'm trying to do. The "symptoms", however, are identical.
For the record let's say that I would like to redirect the output of a local command that needs to be run as root (sudo) to a remote command that needs to be ran as root (sudo). And looking at your last example I see that output redirection per se isn't working with a remote sudo. So this is the issue that I would like to attack. Is there a way to "fix" this?

Is there a way to redirect local command output to a remote sudo command?

Authorize the user to execute that command with no password, and don't require a tty.

In /etc/sudoers on remote.host:
gets:

(edit: note I'm using keys not passwords, yours will still prompt for the remote-host password)

Thank you for your reply.
No password for root is not an acceptable solution. Is there no other way to do this?

Only way I can think of to get it done is to add !tty_tickets,ticket_timeout=0.5 or something short like that to the remote user's sudo options, and do a separate sudo -v beforehand, so for the above substitute

in /etc/sudoers on the remote host:
and the sequence becomes

Got it. Make a named pipe on the remote host and ssh-pipe your data through that, no need for any sudoers tweaking.

That will hang waiting for data, so separately you do
and you'll see on the first one:
The '(cat data | sudo cat)' avoids an annoying delay before sudo prompts for the password (because the redirect tells the shell to open the pipe before running sudo).