I highly suggest a written agreement. There also needs to be an agreement among the members of the pool about separate individual purchases, and if you're the one running the pool, I suggest you NEVER buy separate tickets for yourself, at least not for the game in which the pool is for. For example, if you run a lottery pool, and Powerball is really big, say $300M, and you and 4 other people go $2 in for a $10 ticket, while you're at the store, don't buy a separate Powerball ticket soley for yourself because if that ticket wins, the other people in the pool aren't very likely to believe you when you say "Oh, well that's a separate ticket that I bought for myself, which wasn't included in the pool". When the jackpot gets that big, I could see how someone would be tempted to buy additional separate tickets for themself, but if you're running a lottery pool, doing so would be a bad idea.
Even if you are in a pool and you're not the one purchasing the tickets, I would still be careful about purchasing your own tickets for that game. It comes down to trust. For all you know, the one running the lottery pool(and perhaps someone else in the pool) might lie to the group and say he/she gave you the money to buy the group's tickets. Ya, I know that's a little left field, but money brings out the worst in people. When millions of dollars are at stake, people are willing to do or say anything to get a piece of it.
I would also include in the agreement, that the pool is on a per-drawing basis, which means that if someone doesn't put in money for a particular drawing, they will not be included in on the winnings for that drawing. In the event the pool plays together for months and months and then one day someone misses out on a drawing and the tickets bought for the group wins the jackpot, this will prevent them from using a "we agreed to split it no matter what" or similar argument if they decide to sue the group.