Hi,
Yes, there's a LocalGeometryService in the ESRI.ArcGIS.Client.Local namespace which should be used to provide the URL property to the GeometryService task in the ESRI.ArcGIS.Client.Tasks namespace. For example the C# code below will help you achieve your goal.
            // Create a new LocalGeometryService
            LocalGeometryService localGeometryService = new LocalGeometryService();
            // Start the LocalGeometryService (we'll do it synchronously because it starts quickly)
            localGeometryService.Start();
            // Create a GeometryTask and assign the URL from the LocalGeometryService
            GeometryService geometryTask = new GeometryService()
            {
                Url = localGeometryService.UrlGeometryService
            };
            // Use the Geometry task...
            // e.g. geometryTask.Project(...);
            MapPoint mapPoint = new MapPoint(-0.126, 51.500, new SpatialReference(4326));
            geometryTask.ProjectCompleted += (senderObject, graphicsEventArgs) =>
            {
                Graphic resultGraphic = graphicsEventArgs.Results[0];
                if (resultGraphic.Geometry.Extent != null)
                {
                    resultGraphic.Symbol = new SimpleMarkerSymbol()
                    {
                        Color = new SolidColorBrush(Colors.Red),
                        Size = 14,
                        Style = SimpleMarkerSymbol.SimpleMarkerStyle.Circle
                    };
                    MapPoint resultMapPoint = resultGraphic.Geometry as MapPoint;
                    resultGraphic.Attributes.Add("Output_CoordinateX", resultMapPoint.X);
                    resultGraphic.Attributes.Add("Output_CoordinateY", resultMapPoint.Y);
                    GraphicsLayer graphicsLayer = new GraphicsLayer();
                    graphicsLayer.Graphics.Add(resultGraphic);
                    MapControl.Layers.Add(graphicsLayer);
                    MapControl.PanTo(resultGraphic.Geometry);
                }
                else
                {
                    MessageBox.Show("Invalid input coordinate, unable to project.");
                }
            };
            geometryTask.ProjectAsync(new List<Graphic>() { new Graphic() { Geometry = mapPoint } }, new SpatialReference(3857));
The LocalGeometryServiceTask class is an extension to the GeometryService task and contains two convenience methods for initializing a Geometry task with a LocalGeometryService so in the example above the first few lines could be abbreviated to:
            GeometryService geometryTask = new GeometryService();
            LocalGeometryService localGeometryService = LocalGeometryServiceTask.InitializeWithLocalService(geometryTask);
Many thanks for your feedback.
Cheers
Mike