How does summoning players for bosses affect boss stats?

Solution 1:

Each summoned player, or NPC will fractionally increase the health of the boss. I don't know offhand by how much for each summon. You can have up to three summons. Basically, it's there to make sure you can't cheese the bosses.

A great wiki to use for information about the lore, mechanics, etc. of Dark Souls is the Fextralife wiki. I highly recommend checking it out. Edit: Boss damage, resistances, and weaknesses do not change.

Edit 2: I did some looking around on forums and Reddit, and the general consensus is between 10-20% more health per summon. In Dark Souls 1, health was increased 50% per summon, and in Dark Souls 2, boss defences increased.

Edit 3: To answer the second part of your question, when your summon(s) enter the boss room, their health is automatically increased, just like the enemies outside of the boss rooms. The effect stays until either you die, or defeat the boss, no matter how many summons die.