I believe Bing Maps (BingMapsTileSource) supports SSL, by the way.

Here's how you might proxy the requests for the tile images through a secure site you host.
Before you proceed in this manner, please check to see if this is a valid usage of OpenStreetMap according to their terms of use.
Also, be aware that since you are basically proxying all the OpenStreetMap image traffic for your application, that you would be increasing your site's bandwidth consumption considerably.
Also note, if this is a publically accessible site, you will want to secure access to img.aspx to prevent access to the image proxy from applications other than your own.
Unless you do this you may end up with other people chewing up your bandwidth!

The Xaml:

        <ig:XamMap Name="xamMap1" Loaded="xamMap1_Loaded" >
            <ig:XamMap.MapTileSource>
                <local:ProxiedOpenStreetMapTileSource />
            </ig:XamMap.MapTileSource>
        </ig:XamMap>

The code behind:

    public class ProxiedOpenStreetMapTileSource
        : MapTileSource
    {
        public ProxiedOpenStreetMapTileSource()
            : base(134217728, 134217728, 256, 256, 0)
        {
        }

        protected override void GetTileLayers(
            int tileLevel, 
            int tilePositionX, 
            int tilePositionY, 
            IList<object> tileImageLayerSources)
        {
            int zoom = tileLevel - 8;
            //replace this with the appropriate url for your environment.
            string secureSiteRoot = "https://localhost/SilverlightApplication62.Web/";
            if (zoom > 0)
            {
                string uri = 
                    String.Format(
                    "{0}img.aspx?Z={1}&X={2}&Y={3}",
                    secureSiteRoot,
                    zoom, tilePositionX, tilePositionY);
                tileImageLayerSources.Add(new Uri(uri, UriKind.Absolute));
            }
        }
    }

and the implementation of img.aspx.
Please note that you would probably want to add some sort of caching/security here, etc, this is just a sketch.

public partial class img : System.Web.UI.Page
    {
        protected void Page_Load(object sender, EventArgs e)
        {
            Response.ContentType = "image/png";
            Response.BufferOutput = true;
            Response.Clear();

            int z;
            int y;
            int x;
            if (int.TryParse(Request.QueryString["Z"], out z) &&
                int.TryParse(Request.QueryString["X"], out x) &&
                int.TryParse(Request.QueryString["Y"], out y))
            {
                string url = "http://tile.openstreetmap.org/{Z}/{X}/{Y}.png"
                    .Replace("{Z}", z.ToString())
                    .Replace("{X}", x.ToString())
                    .Replace("{Y}", y.ToString());

                WebRequest req = WebRequest.Create(url);
                WebResponse resp = req.GetResponse();

                resp.GetResponseStream().CopyTo(Response.OutputStream);
            }

        }
    }

Anyway hope that explains how one way you might approach consuming OpenStreetMap or other services from a secure Silverlight application.
But there are additional concerns you would have to address before using something like this in production.

-Graham

Thanks for the advice. I understand what needs to be done now.