First, I thought, unluckily, I hit a bug on Json.NET while serializing enum but after testing the issue I found out that it seems something by design but I could not find it documented. So, here you are, tests I wrote to study the issue.

ReadOnlyPropertiesIssue.cs

//#define UsingStringEnumConverter

using System;
#if UsingStringEnumConverter
using Newtonsoft.Json.Converters;
#endif
using NUnit.Framework;

namespace Newtonsoft.Json.Tests.Converters
{
	class ReadOnlyPropertiesIssue
	{
		static class TestHelper
		{
			public static void AssertCanSerialize<T>(T toSerialize, string expectedJson)
				where T : ITestObject, IEquatable<T>
			{
#if UsingStringEnumConverter
				string json = JsonConvert.SerializeObject(toSerialize, Formatting.Indented, new StringEnumConverter());
#else
				string json = JsonConvert.SerializeObject(toSerialize, Formatting.Indented);
#endif
				Console.WriteLine("Expected: {0} \r\nActual: {1}", expectedJson, json);
				Assert.AreEqual(expectedJson, json);
			}

			public static void AssertCanDeserialize<T>(string toDeserialize, T expectedObject)
				where T : ITestObject, IEquatable<T>
			{
				var obj = JsonConvert.DeserializeObject<T>(toDeserialize);
				Console.WriteLine("Expected: [{0}, \"{1}\", \"{2}\"]", expectedObject.UserID, expectedObject.UserName, expectedObject.UserRole);
				Console.WriteLine("Actual: [{0}, \"{1}\", \"{2}\"]", obj.UserID, obj.UserName, obj.UserRole);
				Assert.IsTrue(expectedObject.Equals(obj));
			}
		}

		public enum UserRoles { Admin = 101, User = 102, Guest = 501 }

		private const int USER_ID = 1001;
		private const string USER_NAME = "Waheed Sayed";
		private const UserRoles USER_ROLE = UserRoles.Guest;

#if UsingStringEnumConverter
		private const string USER_IN_JSON = @"{
  ""UserID"": 1001,
  ""UserName"": ""Waheed Sayed"",
  ""UserRole"": ""Guest""
}";
#else
		private const string USER_IN_JSON = @"{
  ""UserID"": 1001,
  ""UserName"": ""Waheed Sayed"",
  ""UserRole"": 501
}";
#endif

		#region Cases to be tested

		public interface ITestObject
		{
			int UserID { get; }
			string UserName { get; }
			UserRoles UserRole { get; }
		}

		public class ReadWritePropsAndNoDefaultCtor : ITestObject, IEquatable<ReadWritePropsAndNoDefaultCtor>
		{
			public int UserID { get; set; }

			public string UserName { get; set; }

			public UserRoles UserRole { get; set; }

			public ReadWritePropsAndNoDefaultCtor(int id, string name, UserRoles role)
			{
				UserID = id;
				UserName = name;
				UserRole = role;
			}

			public bool Equals(ReadWritePropsAndNoDefaultCtor that)
			{
				return UserID == that.UserID && UserName == that.UserName && UserRole == that.UserRole;
			}
		}

		public class ReadWritePropsAndDefaultCtor : ITestObject, IEquatable<ReadWritePropsAndDefaultCtor>
		{
			public int UserID { get; set; }

			public string UserName { get; set; }

			public UserRoles UserRole { get; set; }

			public ReadWritePropsAndDefaultCtor() { }

			public ReadWritePropsAndDefaultCtor(int id, string name, UserRoles role)
			{
				UserID = id;
				UserName = name;
				UserRole = role;
			}

			public bool Equals(ReadWritePropsAndDefaultCtor that)
			{
				return UserID == that.UserID && UserName == that.UserName && UserRole == that.UserRole;
			}
		}

		public class ReadOnlyPropsAndNoDefaultCtor : ITestObject, IEquatable<ReadOnlyPropsAndNoDefaultCtor>
		{
			[JsonProperty]
			public int UserID { get; private set; }

			[JsonProperty]
			public string UserName { get; private set; }

			[JsonProperty]
			public UserRoles UserRole { get; private set; }

			public ReadOnlyPropsAndNoDefaultCtor(int id, string name, UserRoles role)
			{
				UserID = id;
				UserName = name;
				UserRole = role;
			}

			public bool Equals(ReadOnlyPropsAndNoDefaultCtor that)
			{
				return UserID == that.UserID && UserName == that.UserName && UserRole == that.UserRole;
			}
		}

		public class ReadOnlyPropsAndDefaultCtor : ITestObject, IEquatable<ReadOnlyPropsAndDefaultCtor>
		{
			[JsonProperty]
			public int UserID { get; private set; }

			[JsonProperty]
			public string UserName { get; private set; }

			[JsonProperty]
			public UserRoles UserRole { get; private set; }

			public ReadOnlyPropsAndDefaultCtor() { }

			public ReadOnlyPropsAndDefaultCtor(int id, string name, UserRoles role)
			{
				UserID = id;
				UserName = name;
				UserRole = role;
			}

			public bool Equals(ReadOnlyPropsAndDefaultCtor that)
			{
				return UserID == that.UserID && UserName == that.UserName && UserRole == that.UserRole;
			}
		}

		public class ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor : ITestObject, IEquatable<ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor>
		{
			[JsonProperty("UserID")]
			private readonly int userID;

			[JsonProperty("UserName")]
			private readonly string userName;

			[JsonProperty("UserRole")]
			private readonly UserRoles userRole;

			[JsonIgnore]
			public int UserID
			{
				get { return userID; }
			}

			[JsonIgnore]
			public string UserName
			{
				get { return userName; }
			}

			[JsonIgnore]
			public UserRoles UserRole
			{
				get { return userRole; }
			}

			public ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor(int id, string name, UserRoles role)
			{
				userID = id;
				userName = name;
				userRole = role;
			}

			public bool Equals(ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor that)
			{
				return UserID == that.UserID && UserName == that.UserName && UserRole == that.UserRole;
			}
		}

		#endregion

		#region Case 1

		[Test]
		public void SerializeReadWritePropsAndNoDefaultCtor()
		{
			var obj = new ReadWritePropsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanSerialize(obj, USER_IN_JSON);
		}

		[Test]
		public void DeserializeReadWritePropsAndNoDefaultCtor()
		{
			var obj = new ReadWritePropsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanDeserialize(USER_IN_JSON, obj);
		}

		#endregion

		#region Case 2

		[Test]
		public void SerializeReadWritePropsAndDefaultCtor()
		{
			var obj = new ReadWritePropsAndDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanSerialize(obj, USER_IN_JSON);
		}

		[Test]
		public void DeserializeReadWritePropsAndDefaultCtor()
		{
			var obj = new ReadWritePropsAndDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanDeserialize(USER_IN_JSON, obj);
		}

		#endregion

		#region Case 3

		[Test]
		public void SerializeReadOnlyPropsAndNoDefaultCtor()
		{
			var obj = new ReadOnlyPropsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanSerialize(obj, USER_IN_JSON);
		}

		[Test]
		public void DeserializeReadOnlyPropsAndNoDefaultCtor()
		{
			var obj = new ReadOnlyPropsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanDeserialize(USER_IN_JSON, obj);
		}

		#endregion

		#region Case 4

		[Test]
		public void SerializeReadOnlyPropsAndDefaultCtor()
		{
			var obj = new ReadOnlyPropsAndDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanSerialize(obj, USER_IN_JSON);
		}

		[Test]
		public void DeserializeReadOnlyPropsAndDefaultCtor()
		{
			var obj = new ReadOnlyPropsAndDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanDeserialize(USER_IN_JSON, obj);
		}

		#endregion

		#region Case 5

		[Test]
		public void SerializeReadOnlyPropsWithBackingFieldsAndNoDefaultCtor()
		{
			var obj = new ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanSerialize(obj, USER_IN_JSON);
		}

		[Test]
		public void DeserializeReadOnlyPropsWithBackingFieldsAndNoDefaultCtor()
		{
			var obj = new ReadOnlyPropsWithBackingFieldsAndNoDefaultCtor(USER_ID, USER_NAME, USER_ROLE);
			TestHelper.AssertCanDeserialize(USER_IN_JSON, obj);
		}

		#endregion
	}
}