How to propagate -Verbose to module functions?

The reason this is happening is because the $VerbosePreference is not propagated when the module is called. I modified your script to explicitly print the value at the same points you are outputting via Write-Verbose and Write-Output.

This powershell.org post proposes adding this to the module, which worked like a charm for me:

if (-not $PSBoundParameters.ContainsKey('Verbose'))
{
    $VerbosePreference = $PSCmdlet.GetVariableValue('VerbosePreference')
}

_{One of the comments mentions bug report with link (it doesn't exist or I don't have permissions to view)}

The issue is discussed in a TechNet post, with a link to a Get-CallerPreferance function that addresses this issue.

---

Module:

function Show-VerbosityB { [cmdletbinding()]Param()

    <# uncomment to get verbose preference from caller, when verbose switch not explicitly used.
    if (-not $PSBoundParameters.ContainsKey('Verbose'))
    {
        $VerbosePreference = $PSCmdlet.GetVariableValue('VerbosePreference')
    }
    #>

    Write-Output "`nShow-VerbosityB called"
    Write-output "Global pref: $($global:VerbosePreference)"
    Write-output "Script pref: $($script:VerbosePreference)"
    Write-output "Effect pref: $VerbosePreference"
    Write-Verbose "Show-VerbosityB is Verbose"
}

Caller:

Import-Module C:\Mod

Write-output "On startup: $VerbosePreference"

function Show-VerbosityA { [cmdletbinding()]Param()
  Write-Output "`nShow-VerbosityA called"
  Write-output "Global pref: $($global:VerbosePreference)"
  Write-output "Script pref: $($script:VerbosePreference)"
  Write-output "Effect pref: $VerbosePreference"
  Write-Verbose "Show-VerbosityA is Verbose"
}

function Show-Verbosity { [cmdletbinding()]Param()
  Write-Output "`nShow-Verbosity called"
  Write-output "Global pref: $($global:VerbosePreference)"
  Write-output "Script pref: $($script:VerbosePreference)"
  Write-output "Effect pref: $VerbosePreference"
  Write-Verbose "Show-Verbosity is Verbose"
  Write-Output "`nTesting propagation"
  Show-VerbosityA
  Show-VerbosityB
}

Show-Verbosity -Verbose