Can I see how many bytes are being sent by a particular SyncVar variable?

I think if theres anything to be had its in here.

The maximum number of bytes will not change given the amount of memory required for that data type and the max send interval. Also take in to account that a SyncVar registers changes and does not poll. I do believe that it only likes basic data-types and so will likely not work on a Struct.

There is a SyncListStruct but its not an attribute, its a class.

I tried it with a struct and it worked fine. I passed this struct through.

 public struct PosRot$$anonymous$$odel {
     public Vector3 Pos;
     public Quaternion Rot;
     public double Timestamp;
     public PosRot$$anonymous$$odel(Vector3 Pos, Quaternion Rot, double Timestamp){
         this.Pos = Pos;
         this.Rot = Rot;
         this.Timestamp = Timestamp;
     }
 }

The class script I used was

 using UnityEngine;
 using UnityEngine.Networking;
 using System.Collections;
 
 [NetworkSettings (channel = 0, sendInterval = 0.1f)]
 public class NetworkSyncPosRotStruct : NetworkBehaviour {
     
     [SyncVar (hook = "Sync$$anonymous$$essage")]
     private PosRot$$anonymous$$odel $$anonymous$$essage = new PosRot$$anonymous$$odel();
 
     void FixedUpdate(){
         var message = new PosRot$$anonymous$$odel (this.transform.position, this.transform.rotation, Network.time);
         this.Transmit$$anonymous$$essage (message);
     }
 
     public virtual bool TransmitCondition(){
         return this.isLocalPlayer;
     }
 
     [Command]
     protected virtual void Cmd_Trasmit$$anonymous$$essageFromClientToServer(PosRot$$anonymous$$odel model){
         this.$$anonymous$$essage = model;
     }
     
     [ClientCallback]
     protected virtual void Transmit$$anonymous$$essage(PosRot$$anonymous$$odel model){
         if(this.isLocalPlayer){
             this.Cmd_Trasmit$$anonymous$$essageFromClientToServer(model);
         }
     }
 
     [Client]
     protected virtual void Sync$$anonymous$$essage(PosRot$$anonymous$$odel message){
         this.$$anonymous$$essage = message;
     }
     
     ///@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
     ///@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
     ///@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
     ///@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
 
     public float LerpRate = 5f;
     
     // Update is called once per frame
     void Update () {
         if (!this.isLocalPlayer) {
             this.LerpPositionRotation ();
         }
     }
     
     void LerpPositionRotation(){
         this.transform.position = Vector3.Lerp (this.transform.position, this.$$anonymous$$essage.Pos, LerpRate);
         this.transform.rotation = Quaternion.Lerp (this.transform.rotation, this.$$anonymous$$essage.Rot, LerpRate);
     }
 }

I tried NetworkTransport.GetPacketReceivedRate, but I keep getting 50 whether I have 1 script or 2 sending messages so I figure I'm doing it wrong.

 [Server]
     void Update(){
         byte e;
         int r = NetworkTransport.GetPacketReceivedRate(0,
                                                        1, 
                                                        out e);
         byte e2;
         int r2 = NetworkTransport.GetPacketReceivedRate(0,
                                                         0, 
                                                         out e2);
         byte e3;
         int r3 = NetworkTransport.GetPacketReceivedRate(0,
                                                         2, 
                                                         out e3);
         Debug.Log (r + "-" + e + "\n" + r2 + "-" + e2+ "\n" + r3 + "-" + e3);
     }

I replaced GetPacketReceivedRate with GetPacketSentRate and got 16-20 bytes per second for each script, regardless of whether I was only sending a single int or whether I was sending a struct with a Vector3/Quaternion/double. So I assume that both functions aren't telling me what's going on.

 using UnityEngine;
 using UnityEngine.Networking;
 using System.Collections;
 
 [NetworkSettings (channel = 0, sendInterval = 0.1f)]
 public class NetworkNothing : NetworkBehaviour {
     
     [SyncVar (hook = "SyncInt")]
     private int $$anonymous$$essage;
     
     void FixedUpdate(){
         var message = Random.Range(1,10000);
         this.Transmit$$anonymous$$essage (message);
     }
     
     public virtual bool TransmitCondition(){
         return this.isLocalPlayer;
     }
     
     [Command]
     protected virtual void Cmd_Trasmit$$anonymous$$essageFromClientToServer(int model){
         this.$$anonymous$$essage = model;
     }
     
     [ClientCallback]
     protected virtual void Transmit$$anonymous$$essage(int model){
         if(this.isLocalPlayer){
             this.Cmd_Trasmit$$anonymous$$essageFromClientToServer(model);
         }
     }
     
     [Client]
     protected virtual void SyncInt(int message){
         this.$$anonymous$$essage = message;
     }
 
 }

I don't actually know the answer, but I can make a guess. Floats take up 4 bytes each, and a Vector3 has three floats, for a total of 12 bytes. Strings take up two bytes per character (I believe), which means your example string, with 21 characters, takes up 42 bytes. Unless the Vector3 has much higher overhead, I would expect it to be more efficient.