If the machine originally sold with type designed specifically for it, avoid it like the plague. Otherwise, a practical, serviceable hot-stamping machine will use the same type as any other machine in a letterpress shop—off-the-shelf foundry type.

Yes, my limited experience with these machines suggests they all need brass type, not lead (foundry) type.

Type specifically made for hot foil stamping machines is generally brass or zinc. Although the standard temperatures used are not enough to actually melt metal type, they do soften it a little and cause it to wear faster. I don't know how much faster. For light usage and one-off printing I expect to mostly do, I am in principle willing to try foundry type as well. But I'd rather use the "right stuff" where possible.

For the composition of foundry type, even cold metal type has always been an alloy, and always mostly lead. According to Carter’s A View of Early Typography, Plantin in 1580 used “82 per cent lead, 9 per cent tin, and 6 per cent antimony, with a little copper. The modern standard for a hard typefounder’s alloy is about 15 per cent tin, 25 per cent antimony, and 60 per cent lead with a trace of copper.” However, smaller foundry metal type often had a higher tin content to give it the fluidity necessary to get into small cavities.

Foundry type has always been mostly lead. It may simply have slightly more tin or antimony (or, rarely, a bit of copper) than hot metal type.