[Verizon Galaxy Note 3 Q&A, Help & Troubleshooting] IT_Architect:
I have figured out my issues, but more importantly I am figuring out how things work. This issue is not specific to any particular Samsung phone or Android version.

Samsung Messages app and requirements

- The Messages app is the default Samsung messaging app. It is designed to work with the default Samsung Contacts app, which has the ability to define a "Message tone"(S5) or "Message alert"(Note 3) by contact. The property may already display in Contacts, or you may have to press a button in Contacts to add the field. Then simply specify the alert/tone you want.

- The contact must be a DEVICE Contact. For example, in my case I use Akrutosync to sync my Outlook with my phone. This requires setting up an Active Sync. Outlook and the phone sync perfectly, however any contact synced like this will only play the default Message tone/Message alert. If you go into the contact, and look at the Connection property, it will show Microsoft Exchange Active Sync. In order for it work work, it must say Device.

- The contact must have a phone number, and that number must be the source of the incoming message. For myself in the IT industry, it is commonplace to have server monitors send a text message via ###-###-####@vtext.com from an e-mail address. We want a different tone/alert for a server down than normal texts. The messages will show as coming from the proper contact, but it will only trigger the default Message tone/Message alert. Samsung said it shouldn't work that way, but I don't have a work-around for the problem.

Verizon Messages+ app and requirements

- Verizon Messages+ app is the default messaging app for Verizon phones. Messages+ can use Verizon's cloud if desired and your account supports it. That makes it simpler to create and respond to text messages for a specific phone number from multiple devices, including computers and tablets. It also means Verizon has access to your personal correspondence. However, even as a local application, It has built-in functionality to record an MMS message, send postcards, pictures from storage or camera, add captions to pictures, markup pictures, make collages, send your location with the click of a button, activate Glympse so people can follow you or link up with you, send someone a contact, send an MP3 or WAV file.

- Messages+ is designed to work with the default Samsung Contacts app, but DOES NOT USE the "Message tone"(S5) / "Message alert"(Note 3) property in Contacts.

- There is no way to do a contact-specific tone/alert in Message+. The best you can do is a conversation-specific tone/alert, and never delete the entire conversation. To do a conversation-specific tone/alert, at the top left next to the contact's picture, there are 3 horizontal lines. Flick those to the right to open a new window. Scroll to "Customize conversation", and then select Tones where you can select your tone for the conversation. Of course if you ever delete the conversation, the tone you set up will be lost also, and the next time you receive a message from the same contact, you will receive the default tone/alert. Message+ does not make use of the Message alert/tone set up in Contacts.

Note: Even when a 3rd party messaging app is set up correctly, you may still get the default tone. You may need to go to the main settings menu, pick 'Sound' then scroll down to 'Messages' under Samsung applications and turn that off.

Let me know if this helps you with your situation.