So, apparently due to the changes being done to the libraries, every now and then it will stop working. So this is what helped me to make it work in April 2019. Hopefully it will help others.
I haven't added any log or anything specific to my function app, right now for me it's pretty much printing all the console data to the Trace logs in my app insights (locally, not needing yet to configure anything in Azure). Also, my test scenario is running on TargetFramework netstandard2.0 and AzureFunctionsVersion 2.
First you need to add this NuGet package to the function app: https://www.nuget.org/packages/Microsoft.Azure.WebJobs.Logging.ApplicationInsights/
Then you have to add an empty file to your project root: ApplicationInsights.config you don't need to copy it to output when building or anything, this is just something that has to be there or Application Insights won't work locally.
Then you need to add some value to your APPINSIGHTS_INSTRUMENTATIONKEY in local.settings.json. This is where it differs from regular console apps, where even if the instrumentation key is empty it will work locally. In function apps apparently you need to add some value there. It can be anything, I've done this and it worked fine:
"APPINSIGHTS_INSTRUMENTATIONKEY": "you-need-to-have-anything-at-all-here-so-it-will-work-locally"