fde101

Drop the shape layer on top on the image layer’s thumbnail, not to the right of it.
That’s exactly what I did, and what I meant by “dragging a shape to the masking/cropping position for the image layer and applying a Gaussian Blur layer effect [to the shape layer] to soften the edges”.

ok, took me another look to catch on... @Alfred and @Michail are blurring the mask rather than the image, and you can certainly do that in Publisher, as indicated.  This will give a fuzzy shape to the edges but the picture itself will remain sharp and crisp.
If you want to blur the actual image, you will need to use the live filter instead, and only Photo can add those.
Looking more closely, the OP's examples do seem to be blurred masks, so this technique is probably more relevant than what I described.

You can feather the edges using the transparency tool, depending on the shape..
I'm not really seeing a way to blur just the edges using Publisher, but if you also have Photo, you can prepare such a shape using Photo, copy/paste it into Publisher, then save it into the Assets panel as an asset that can be reused.
 
As an example:
Create the desired shape using a shape tool or any other means of creating a vector shape within Photo. Duplicate the layer; leave it precisely over top of the original layer. Give one of the layers a fill but no stroke. Give the other layer a stroke but no fill; make the stroke thick (70pts or so) and make sure it is center aligned (the leftmost "Align" option in the Stroke popup from the context toolbar, which is the default) Set the stroke of the stroked object to an elliptical gradient; reduce the opacity of the point that shows up on the right in the popup to around 50% or perhaps just over. With the filled object selected, add a Gaussian Blur live filter using the popup menu on the Layers panel (this is the one step you can't do in Publisher or Designer - only Photo can create these, though they do get copied to the other programs along with the final object).  Set the radius to around 9 or so. In the layers palette, drag the stroked shape layer to the right edge of the thumbnail of the blur filter to turn it into a mask for the filter (it should have a crop icon over it in the palette after you place it there) This should leave you with the filled-shape layer containing the blur filter containing as a mask the stroked shape layer. Select the filled-shape layer and Edit/Copy Switch to Publisher and Edit/Paste Drag the shape in Publisher into an appropriate place in the Assets panel (create a new category for these first)  
To use this:
Drag the shape out of the Assets panel (after the first one you already pasted into the document from Photo) Expand the filled shape in the layers panel, but leave the blur filter layer collapsed so that the mask inside of it is not showing Drag the image layer containing the photo underneath the blur filter layer, but as a sibling not a child (it should be aligned with the blur filter but a child of the solid shape)  
You might need to experiment after that with the thickness of the stroke, the blur radius, and so on, but you can make those adjustments within Publisher once the filter has been created.  You can also do these after the first one from within Publisher, without having to switch to Photo, by copying the blur layer from another shape you already created.

It was explained early in the beta process that there are two types of documents now: artboard-based (Photo) and spread-based (Publisher) which all three applications will be able to open, though possibly not create.  Spread-based documents will not support artboards and artboard-based documents will not support multiple pages.
The impression I got, though not really spelled out, was that Photo will only create artboard-based documents, Publisher will only create spread-based documents, and Designer will have the option of creating either one.  I could be mistaken, but that is how I interpreted what was being said by the Serif team early on.

I don't know, but it seems odd to me that they aren't there in the beta, certainly for Designer. We still need a page selector in both Affinity Photo and Affinity Designer so multipage documents are managable in the other two as round tripping may add further pages.

I agree, and I generally do this 90% of the time, as I can never be sure either when someone else might post in between...
 
You can quote just a few words as in this example, to help maintain the flow of the conversation.
 
Alternatively, you could at least provide the name of the person you are responding to (or even just "OP" in this case would have helped), as that would also make it more clear what is going on.  You did post about 1/2 hour after @Puck did, so I probably jumped to the conclusion as well that you were replying to him rather than to @Husbandman the first time I read through that.

I agree! But in THIS case: Read my first comment and tell me, @fde101, if its unclear, which post it refers to. (Or better: Don‘t tell me, in order to not confuse this thread even more.)

I think a better question is: Will Photo and Designer get an "Edit in Publisher" menu item?
Without that missing function there seems to be no good way to get back to Publisher after you've used "Edit in Photo" or "Edit in Designer".

Master A layer was sitting in the middle of a group of layers. The moment I selected the group, I would automatically get that master layer.
I thing the true bug is that we can select & group the master layer with other object-layers. If they make it non-selectable and always at the bottom, bellow all the other layers, this wouldn't happen.

JPEG does not support transparency so you will need to use a different format (PNG, TIFF, etc.)

You can drag guides out from the rulers when using the move tool.

Of course. that is, what @Joachim_L already stated.

What if you select three or more text frames? Do you want them all to have text rulers showing, or do you only want the rulers showing when you are working in a text frame? I prefer to have as little clutter on the page as possible.

We can ask to get something implemented, but not want 
I have used vertical selections in another program but that was a text editor with a fixed width font. I see big problems for vertical selection with variable width fonts because it will be very hard to figure out what has to be part of the selection and what not.
d.

An entire new application is being added to the suite with the 1.7 update (Publisher) and there are quite a few things that need to be worked out because of that.  It is understandable that this one will take a bit longer than usual.

You can drag guides out from the rulers when using the move tool.

An entire new application is being added to the suite with the 1.7 update (Publisher) and there are quite a few things that need to be worked out because of that.  It is understandable that this one will take a bit longer than usual.

Humans are a niche market compared to cockroaches.  As cockroaches are unlikely to buy these products due to lack of an appropriate language translation I suppose they should just give up and stop making them?
The fact that Windows continues to outsell the MacOS seems to be evidence that most people in the world are willing to put up with inferior products.  Either that or it simply takes multiple Windows systems to do the work of one Mac.

it's a bug and has been reported

My late brother-in-law and I could argue for hours over whether that is a typo or a spelling mistake. Good times, my sister will disagree about the 'Good' as she had to listen to it.

Bear in mind that the three Affinity applications use the same file format, so from that perspective Photo = Designer = Publisher and placing a document from Affinity Photo is no different from placing one for Affinity Publisher...

Sounds like you are talking about completely different things.

I propose that the menu option be called "Perform Evil Export..."
That way all of the problems that come up in the exported file should be excused since the export was evil to begin with and thus cannot be trusted.

done

That's a better question for a separate topic in Discussions and Suggestions for Affinity Publisher Beta on Desktop, in my opinion.