loose prop

Many possible causes and solutions; Try dry fitting without the keys and marking how far the wheel goes up on the taper (without torqueing or using the nuts, just slip them up), then installing the keys (making sure they don't slide up too far in the keyway), the wheels should go back to the marks, if not the key's are binding them. Second possibility, put on and torque the smaller nut first followed by the larger one, doubt it will make a difference but worth a try. Third is to get a Propsmith or similar device to press them in place (and remove them with less struggle!). I have seen yards apply a little goo (4200 or 5200 sealant) on the threads before putting the nuts on but personally wouldn't do that, the next guy who has to remove them won't be calling you "honey"!

Smaller nut goes on first. I always lap the props when I put them on, Put them on with the nut a bit loose and spin them with valve grinding compound inside the hubs, pull them back off, clean everything, re-install keyway, put props on, tighten nuts. I've NEVER heard of them loosening because of too much torque in reverse, the keyway should keep any movement from loosening the nuts. Have also seen people put 4200 or 5200 on the threads before tightening the nuts, we never did, but have seen it done. Also sometimes the keyway extends too far into the taper of the keyway slot and doesn't allow the props to go all of the way on, Have had to shorten the keyway before as well on some new boats from a quality manufacturer. You should see this if you mark the shafts as Bill described.

I doubt it, but you never know. Big nut/small nut you still have 2 nuts. Just guessing, I think the Keyways might be too long and the nut drives them up the taper, not allowing the props to seat themselves all of the way up the shaft. I would, lap the props, clean all of the valve grinding compound out, put the prop back on without the keyway, snug 1 nut and mark where the prop ends up on the shaft with a magic marker, pull it off and then try it with the keyway in and see where it ends up. Also how tight does the diameter of the keyway fit into the prop and shaft.

In addition to lapping the props with grinding compound you should after cleaning it all off apply Prussian Blue to the cone part of the shaft evenly and slide the prop home to where it would normally sit, once removed any bright bits will be where the prop contacts the shaft.

A trick or two on K1W1 comment as to checking with Prussian Blue:
1. Line the shaft so the keyways are horizontal, not vertical either up or down, but it does not have to be perfectly horizontal.
2. Take out the keyways and clean any noticeable burrs on the keys, and key slots in both the prop and the on the shaft... AND CLEAN PERFECTLY CLEAN THE SHAFT AND PROP MATTING SURFACES... use some light scotchbrite or other to make sure of no rust or anything but bright clean metal.
3. Apply the Prussian Blue in a light even coating to the shaft... important it is thin and even as possible... this take time to learn but a linen or closely woven cotton (like bed sheet scrap) over a neatly folded rag pre-saturated with the blue (exactly like an ink stamp) with no globs on the saturated smooth sheet surface works or buy an old fashioned ink stamp pad and remove it from being glued down in the tin (Store in tin). Apply to shaft by putting a small glob on the smooth surface of you bluing pad (home made or store bought as I said). Using the pad wipe it on the shaft you will find stippling it on helps evenly get it on... then using wide smooth strokes smooth the thin film of blue on the shaft. TO GET EVEN THIN FILM ON SHAFT.
4. Slide the cleaned taper on the prop up the shaft by letting it rest on the top or upper surface of the shaft; this will leave a bright line on the shaft wiping off the bluing. When in place snug your nut lightly. Carefully with a Soft faced hammer (lead preferred) tap the prop to free it. Being careful not to move the prop other than to free from the taper. Carefully slide the prop off the shaft with it resting on the top surface, reverse of assembly.
5. Disregard in "reading the bluing" the line of assembly bring mark.
6. The high spots will show with clean or bright surface.
7. These should be smoothed down with a fine small file a Swiss Crochet files work best.
8. Now some comment about the fit of the taper. The small areas you file even if below the surface and not contacting will make little difference in the holding strength of the taper... there is a large area of contact. That is the thing to remember the taper works by having a large contract area and the slope causes the surfaces to closely mate up. If you only see points of contact those must be worked down to get the overall even close MATING of the surfaces. Remember a few or even many small low or no contact areas like due to pitting will be inconsequential.
9. Importantly, the key is not the primary part holding the prop from turning on the shaft. The fit of the taper does that without the key. The key is really just there as a backup due to imperfections of the fit. With proper fit the shaft and prop are literally like welded together.

If you wonder I learned these things as a kid... I attended a military school for a time and they had a machine shop class taught by a semi-retired German immigrant (full master machinist and tool maker in the guild traditions) instructor. I might say NO ONE got a top mark in that class as his belief was that was reserved for perfection. And, secondly I suppose it was about 1960 and my grandfather had his boat hauled and I remember the old guy English guy doing the prop fitting to the shaft in the yard... in those days insurance protections did not include keeping the client or even interested kids away from the work.

Are the props loose as well or just the nuts?

They shouldn't. The Keyway keeps the props locked to the shaft so they can't turn except with the shaft.

my Viking would always have one prop that was loose on the shaft no matter how tight I made the nuts.
I don't remember which one it was, but every time I was in the water under the boat there was always movement with the prop on the shaft.

The other one required a great deal of force with a prop remover to take off.

just to add I remember the nuts being loose and up against cotter pin a few times.

If you use a propsmith to install the wheels and you have lapped them and previously "blue" tested the taper they should not come loose; the nuts or the props.