Hey StCroixer – You can fish either shore with a MN license. Here is the info from Page 43, 2006 MN Fishing Regulations:
Minnesota has a reciprocal agreement with each of its bordering states: Wisconsin, Iowa, South Dakota, and North Dakota. Residents of Minnesota or a bordering state may fish throughout the waters bordering the two states only if they possess a valid resident license from their resident state. Lake Superior is not covered under this agreement (see page 22). Nonresident anglers who have a nonresident license from either Minnesota or the bordering state may also fish throughout the border waters between the two states. Anglers may launch and fish from either shore and may transport their catch by the most direct route to the state in which they are licensed. This includes children who are not required to have a license.
Quite a few years ago I walked into the Swing Bridge and fished it from the shore. Now it is posted with No Trespassing Signs by the Railroad. You could probably still do it but it is a long hike and the fishing was not that good.
If you are looking for a shorefishing spot I agree with SteveO – the Lake Mallalieu Dam in Hudson is easier to reach and you would probably have just as good luck there. Get yourself the Hudson Map and it will show you how to find the dam. You can drive right up to it and have just a short walk to the open water. I fish there sometimes during the January and February thaws when I want to fish a long rod.