My opinion, absolutely unsafe with modern loads.  Moderately safe with true blackpowder (no substitutes, not smokeless equivalent) loads.  Keep the bullet weight down to keep pressures down with blackpowder.

Could be safe with very carefully designed, low pressure smokeless loads.   

If the hammer in a 44 is of the lugged variety the gun is a smidgen safer, but still not one in which I would fire "modern" loads. Will it "hand grenade" on you? Probably not, but the action pins will crap out in no-time.

Anecdotally, the 417 Walnut Hill was the acme of the 44 design and when Stevens introduced it in .22 Hornet it didn't last long before they realized their mistake and yanked it out of production. And that was with a tiny surface area of the case rim nowhere near the size of all other centerfire cartridges at the time - surface area being the determiner of case thrust against the breech. 

My two centerfire 44's are a .22WCF which I load to 1400fps with a 45 grain bullet and Unique powder, and a .32 Colt Long (with a .32 rimfire chamber + centerfire breech block) that gets 1000fps with an 85 grain heeled bullet + 1.8gr. Bullseye. The rest of my 44's are .22 rimfires and I have zero intention of changing that.

Pictures of your rifle would help identify what you have. Model 44's had quite a few improvements in the early years, such as pins rather than screws for the breech block and lever, center extractor over the early side extractor, and the hammer lug as mentioned. You didn't say whether the action breeches up tightly or if it has the dreaded lever droop.  Also there is a lot of confusion between identifying Model 44's and 44 1/2's, with the 44 1/2 being a true falling block and a much stronger action. With all of that said, if you have a model 44 that the breech block cams up tightly, I would also agree to occasionally shooting it with black powder or very light smokeless loads.