EDIT: After I wrote the following paragraphs, I realized that there is a more obvious way to see that $n-1$ transactions suffice. The proof is by induction and the base case $n=1$ is trivial. If $n-1$ transactions suffice for $n$ people, then given $n+1$ people, the first person can pick any other person, and by making a transaction with that person, either pay the total of what he owes or be paid the total of what he is owed, as appropriate. Then the inductive hypothesis says that the other $n$ people can settle with $n-1$ transactions, for a total of $n$ transactions.
Original answer: Let $n$ be the number of people. Then $n-1$ is an upper bound on the numbers of transactions required to settle all debts, and in the "typical" case no smaller number of transactions suffices. To see this, we may assume that the people numbered $1,\ldots,k$ are net debtors and the people numbered $k+1,\ldots,n$ are net creditors. Let $K$ be the sum of the net liabilities of the people numbered $1,\ldots,k$. We may assume that $K$ is an integer by working in terms of the least monetary unit.
First we choose a partition $P_L$ of $K$ into $k$ pieces, each piece corresponding to the net liability of one of the people numbered $1,\ldots,k$. Next we choose a second partition $P_A$ of $K$ into $N-k$ pieces, each piece corresponding to the net assets of one of the people numbered $k+1,\ldots,n$. The common refinement $P$ generated by $P_L$ and $P_A$ has at most $n-1$ pieces, and each piece corresponds to a transfer from one of the people numbered $1,\ldots,k$ to one of the people numbered $k+1,\ldots,n$.
Moreover, any optimal way of settling the debt does not involve transfers to net debtors or from net creditors, so it arises from some such pair of partitions $(P_L,P_A)$. The choices of $P_L$ and $P_A$ (really just the choice of $P_L$ given $P_A$ or vice versa) may affect the number of pieces in the common refinement $P$ that they generate. However, if we assume that the amount of each debt is chosen randomly, then for a fixed $n$, as $K$ grows the probability that some choice of partitions makes $|P| < n-1$ approaches zero. (This only happens when the total debts of some proper subset of the debtors sum to exactly the same amount as the total assets of some proper subset of the creditors.)