I would write a custom inspector for this problem. To start off you would write your normal Monobehavior script like the following and attach it to a gameObject where you will be able to see the public property visible. But when you get/set the value the Debug.Logs inside Setter/Getter would not print.

using UnityEngine;    
using System.Collections;

public class UseGetterSetter : MonoBehaviour 
{
 // This will be displayed in the inspector as usual
 public int publicVariable;

 // public property of the variable
 public int PublicVariableProperty
 {
 get 
 { 
 Debug.Log("Getter!");
 return publicVariable; 
 }
 set 
 { 
 Debug.Log("Setter!");
 publicVariable = value; 
 }
 }
}

Now you have to create a custom inspector like such to take advantage of your getter/setter.

using UnityEngine;
using UnityEditor;

[CustomEditor(typeof(UseGetterSetter))]
public class CustomInspector : Editor {
 public override void OnInspectorGUI () 
 {
 base.OnInspectorGUI();

 // Take out this if statement to set the value using setter when ever you change it in the inspector.
 // But then it gets called a couple of times when ever inspector updates
 // By having a button, you can control when the value goes through the setter and getter, your self.
        if (GUILayout.Button("Use setters/getters"))
 {
 if(target.GetType() == typeof(UseGetterSetter))
 {
 UseGetterSetter getterSetter = (UseGetterSetter)target;
 getterSetter.PublicVariableProperty = getterSetter.publicVariable;
 Debug.Log(getterSetter.PublicVariableProperty);
 }
 }
 }
}

like I explained in the comment, you could take out the 'button' line in the inspector code (if (GUILayout.Button("Use setters/getters"))) so that what ever you do in the editor to that variable will be reflected back on it through our getter/setter. But it gets called many times. If you don't wish that. You could simply set the value and press the button when ever you want to take advantage of using your getter/setter.