How do you show the other side in negotiations that you're listening to them? Does it give you leverage?

1. Aallow your opponent to speak, while you actively listen.
2. You ask open questions about what they are saying Whydid this happen?, What does this mean?, When did it happen?, Can you explain? etc.
3. The more you engage and the more they can talk the greater the opportunity to build rapport.
4. Your body language has to support your interest.
5. Build empathy by recognising the extend or impact of problem on them.
6. Recognise and use any emotions displayed by opponent wisely, dont ignore them.
7. Once you have built rapport, shown empathy and built trust you can begin to influence (leverage).

How to show Listening?

 1- Maintain eye contact without staring or glaring

 2- Smiling.

 3- Relaxing body language.

 4-Conceal nervousness.

 5- Starting a conversation.

 6- Respond to the talk by asking questions about the conversation.

 7- Avoid probing.

 9- Avoid deflection.

 - Avoid Advising.

 Does it give me leverage?

Absolutely, as I believe effective leadership is effective listening

first of all, you must not interupt your interlocutor and listen carfuly to he or she is saying. second you have to show you enthusiastic and use body language through nod (move the head up and down) in order to indicate your agreement or assent.Finaly, you should express gratitude to your negociator.

Simply, Give the chance to your body to express that

eye contact, free face expressions and smiling will do That, just feel free and listen by heart  

Eye contact is he magic of listening.  If you are listening with your face looking at something else it is sometimes taken as dis-respect.  Your gesture and emotions should be totally focused to the person in-front of you to show your respect and willing to go further in this negotiations.