I hope this is in scope of this stackexchange. I was wondering, if the chat history is saved servers-side (https://msdn.microsoft.com/en-us/library/office/hh243707%28v=office.14%29.aspx), is it then additionally saved locally, on the phone?
The conversation history in Lync Server 2013 was device-specified. It was stored in your Lync 2013 client, and in an Outlook folder labeled "Conversation History", depending on which device you used for that conversation. Example: You used a Laptop, the "Conversation History" folder was used. You used a phone, it got logged on on the phone Lync client.
Now with Skype for Business this was improved. Now all conversation history is stored server-side. Which means that all your devices can see the same Conversation History. BUT you need Exchange Server 2013/2016 or Exchange Online and also need to enable this feature via powershell:
Get-CsClientPolicy -Identity Global |Set-CsClientPolicy -EnableServerConversationHistory $true