There are lapidary materials that can be cut without using specialized saws. Small raw gemstones are not good candidates, however.
Prior to modern lapidary equipment, cutting small gemstones relied on understanding, and taking advantage of, a stones natural cleavage. This method gives very little flexibility as to how and where you cut, and will result in a lot of lost material.
Tile saws, dremels, and end-cut bolt cutters can be used to cut gemstone material without the cost of specialized equipment, but all suffer from limitations.
End-cut bolt cutters: ...can, in a pinch (no pun intended!), cut slabs (sections of gemstone material pre-cut into relatively even thickness slices) to preform for cabochons, but they are most likely to just shatter anything that isn't a flat slab.
Dremels can be fitted with diamond blades to cut gemstone materials with finer precision, but they still have some serious downsides. The biggest concern is health: cutting stone materials generates dust, which if inhaled, can lead to silicosis, a dangerous and debilitating lung disease. Gem cutters typically mitigate this risk by always keeping material wet while cutting and polishing it. This keeps the dust within the water, and out of the air.
Dremels may also have difficulty cutting tougher materials, as unless you obtain a variable-speed tool (which tend to be more expensive), it will likely be too fast. Lapidary saws are designed to cut at a slower speed, to gradually wear away the material without cracking or shattering it, and to keep the blade from binding.
Tile saws also spin too fast for harder lapidary materials. They also tend to have a much larger kerf.
The kerf is the width of the actual cut, where the blade removed materials:
The larger the kerf, the more material is destroyed by the cutting process. When dealing with larger, inexpensive material, this may not be a big problem, but for small gemstones, this is likely to be a significant concern.
Ultimately, your best bet is to use specialized saws. Beginner hobby saws can be relatively affordable, especially if you can obtain one used. If there is a gem and mineral club nearby that you can join, some of them will have club machines available for members to use. Even if the club doesn't own machines, members might be willing to help you out. Just remember that the blades for the machines are expensive, and wear out relatively quickly. Offering some money to cover wear-and-tear would probably be appropriate.