AzureFunctionsのlocal.settings.jsonファイルといえば

 {  
   "IsEncrypted": false,  
   "Values": {  
     "FUNCTIONS_WORKER_RUNTIME": "<language worker>",  
     "AzureWebJobsStorage": "<connection-string>",  
     "AzureWebJobsDashboard": "<connection-string>",  
     "MyBindingConnection": "<binding-connection-string>"  
   },  
   "Host": {  
     "LocalHttpPort": 7071,  
     "CORS": "*"  
   },  
   "ConnectionStrings": {  
     "SQLConnectionString": "<sqlclient-connection-string>"  
   }  
 }  

のような感じで設定情報が保存されてます。