Hi,
I know you only mention an iPhone at your end but you have posted in a Mac area (Mavericks) which implies a Mac as well.
IF there are two devices most people will sync the Contact App List across the two devices.
This is done in System Preferences > iCloud (If you use iCloud) and enabling the Contacts box (or in System Preferences > Internet Account and the account of your choice)
On the iPhone it is Settings > Mail, Calendars and Contacts and then the Contacts section and choosing the account to sync with.
This makes sure the iPhone Numbers and Apple ID for each person can be accessed by the Messages app on each and keep linked IDs together as it were.
Of course it does require that the iPhone has either WiFi or a good enough Carrier Service to stay connected to the iMessages servers.
I work in a place that has a warehouse about 100meters on each side. It has several wifi points but coverage is still somewhat spotty. The Carrier service is also spotty but not always in the same place although there are "dead" spots that include both.
This might explain why his (or her) iPhone may not get some of the iMessages.
They also needs to be syncing the two IDs at their end.
Until recently I was also having trouble with my iMac not showing the iMessages conversations using iPhone Number to iPhone Number that I was having from work to say overtime was happening.
Changing the connection to starting it on my iPad to her Apple ID strangely seems to have sorted this out.
I have my Mac set to "Send from" my Apple ID in new messages. So is the iPad. The iPhone is set to use the iPhone Number.
However in most cases I keep the chats/conversations with family members on show in all version of the app.
As your friend says they see 3 people in the chat you will probably need to check what is happening at your end.
If you are Dismissing Chats and using the New Messages icon and adding their Apple ID and then iPhone Number the app might presume you are restarting a Group chat that used to exist and doing an auto complete that you are not expecting.
If you only use the X when you mouse of the Side Bar you are not deleting the chat you are only removing it from sight (Dismissing it).
Therefore when you start a New Messages it will bring up the "history" of the old chat.
iMessages or in fact the whole app is notoriously bad at keeping IDs and Numbers it thinks it has used in the past even if you remove them from the Address Card.
It is not until you get to Yosemite and iOS 8 that you can do Text Forwarding (SMS) between the Mac and iPhone.
So.
1) The Contacts app should be synced to allow the correct use of only one ID for a person.
2) However that does need them to be Online at all times with their Devices.
3) That requires that they are syncing their ID
4) And that, in some cases, requires that a device is able to get iMessages.
5) Starting a New Chat suggests the previous is not in the Side Bar.
How it was removed from view may be playing a part.
6) Yosemite and iOS 8 might give you options to Text Foreword (Use SMS) across both devices and allow mixed iMessages and SMS conversations (reports seem to vary as to how well this works).
Basically there are some things that can be checked at your end but it is likely that they have not linked their IDs (iPhone and Apple ID) - or they have lost the sync - and this is why they are not contactable with just one ID from your end.
It seems possible that your end is also auto completing when you are not expecting it to.
It seems you have a mixture of possible issues across both ends.

10:02 PM Friday; June 3, 2016
iMac 2.5Ghz i5 2011 (El Capitan 10.11.x)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Sometimes Another OS X version boot
iPhones and an iPad 2