LocalWorld.cs Snippet: Created by another team member, I took their code and changed it to work in a network environment, sending the emotes between clients through the server.

LocalWorld.cs

void Start () {
		emoteDisplayer = new EmoteDisplayClass();
    }

public void displayEmote(int emoteType, NetworkInstanceId netId)
	{
		emoteDisplayer.displayEmote (emoteType, netId);
	}

public class EmoteDisplayClass : ScriptableObject
{
	public List<EmoteButton> buttons;
	private Vector2 Mouseposition;
	private Vector2 fromVector2M;
	private Vector2 centercirlce;
	private Vector2 toVector2M;
	public Sprite sprite1;
	public Sprite sprite2;
	public Sprite sprite3;
	public Sprite sprite4;
	public float EmoteLifetime;
	private bool menuon;
	public int menuItems;

	public void Awake()
	{
		sprite1 = Resources.Load<Sprite>("Sprites/Emotes/HappySprite");
		sprite2 = Resources.Load<Sprite>("Sprites/Emotes/SadSprite");
		sprite3 = Resources.Load<Sprite>("Sprites/Emotes/HelloSprite");
		sprite4 = Resources.Load<Sprite>("Sprites/Emotes/EmergencySprite");
		EmoteLifetime = 2;
		centercirlce = new Vector2(0.5f, 0.5f);
		fromVector2M = new Vector2(0.5f, 1.0f);
		buttons = new List<EmoteButton>();
	}

	public void displayEmote(int emoteType, NetworkInstanceId netId)
	{
		GameObject Player = ClientScene.FindLocalObject(netId);
		//int InstanceID = Player.GetComponent (NetworkInstanceId);
		//string netID = Player.GetComponent<NetworkIdentity>().netId.ToString();
		Debug.Log ("Player network ID Received by client: " + Player.GetComponent<NetworkIdentity>().netId.ToString());
		Vector3 offset = new Vector3(0.0f, 2.5f, 0.0f);

		if (emoteType == 0)
		{
			GameObject emoteobject = new GameObject("Emote");
			SpriteRenderer renderer = emoteobject.AddComponent<SpriteRenderer>();
			IconObject qwe = emoteobject.AddComponent<IconObject>();
			qwe.PlayerNetID (netId);
			renderer.sprite = sprite1;
			Destroy(emoteobject, EmoteLifetime);
		}
		else if (emoteType == 1)
		{
			GameObject emoteobject = new GameObject("Emote");
			SpriteRenderer renderer = emoteobject.AddComponent<SpriteRenderer>();
			IconObject qwe = emoteobject.AddComponent<IconObject>();
			qwe.PlayerNetID (netId);
			renderer.sprite = sprite2;
			Destroy(emoteobject, EmoteLifetime);
		}
		else if (emoteType == 2)
		{
			GameObject emoteobject = new GameObject("Emote");
			SpriteRenderer renderer = emoteobject.AddComponent<SpriteRenderer>();
			IconObject qwe = emoteobject.AddComponent<IconObject>();
			qwe.PlayerNetID (netId);
			renderer.sprite = sprite3;
			Destroy(emoteobject, EmoteLifetime);
		}
		else if (emoteType == 3)
		{
			GameObject emoteobject = new GameObject("Emote");
			SpriteRenderer renderer = emoteobject.AddComponent<SpriteRenderer>();
			IconObject qwe = emoteobject.AddComponent<IconObject>();
			qwe.PlayerNetID (netId);
			renderer.sprite = sprite4;
			Destroy(emoteobject, EmoteLifetime);
		}
	}
}