So are all clients sending "broadcasts" (Sending the data in every direction)?
You clarified in your comments that you don't actually mean "broadcasts" but rather that you are asking if clients transmit omnidirectionally.
The short answer is that most 802.11 client devices transmit in a pattern that covers a complete 360 degrees horizontally around the device because most are mobile and won't have a fixed facing towards the access point. They do this by using dipole antennas.
The characteristics of how any RF device transmits is not determined by the client, but by the antenna used to transmit. Changing the antenna can have a significant impact on how the device transmits a signal.
Generally speaking, most antennas will transmit some signal in most directions, but depending on the actual characteristics of the antenna in question, that signal transmitted in a certain direction can be very low to the point of being entirely useless. The majority of antennas will have a main "lobe" along which the signal will be strongest, a number of side or back lobes with weaker signal, and a number of areas where very little to no signal is generated.
Additionally, with beamforming in use, a transmitter can also influence the nature of the propagated signal by transmitting specially formulated and timed transmissions from multiple antennas to strengthen the signal in certain directions and/or weaken it in others.
Once a RF signal is transmitted, it is then subject to the physics of RF signals, which can bend and/or be reflected so signal will often cover areas where the antenna didn't transmit initially.
WLAN-Clients know when to send because they are listening before talking. In many descriptions I've read that they "hear into the router", if some other client is already transmitting his data.
This is the first time I have come across this description of a client ability to "hear into the router" but wherever you may have come across this, it is highly inaccurate. Clients can only hear what is transmitted in the air.
Now clients do negotiate timing with the access point and the 802.11 network can use a number of control mechanisms to help manage who talks when, but this does not give the client the ability to "hear into the router" in any way.