Can I design PAGX layouts shared in ArcGIS Online in VSP Designer with ArcGIS Pro Layout Mode active?

No, this is currently not possible. VSP Designer can only access layout templates from the Portal which is hosting the VSP Engine Tools. VSP Designer can connect to ArcGIS Online, but the VSP Engine Tools can NOT be hosted by ArcGIS Online due to limitations by ESRI. Because of these limitations, accessing layout templates shared in ArcGIS Online is not possible in VSP Designer in ArcGIS Pro Layout Mode.