Label the people $a_1, a_2 \dots a_8.$ Suppose $a_1$ has zero coins. Then $a_2$ and $a_8$'s coins must sum to an odd number. Thus one of them has to have an even number of coins and the other has to have an odd number. WLOG let $a_2$ have an even number of coins and $a_8$ have an odd number. Now since $a_1$ and $a_2$ both have an even number of coins, $a_3$ has to have an odd number of coins. Since $a_2$ and $a_3$ have an even and odd number of coins respectively, $a_4$ has to have an even number of coins. Since $a_3$ and $a_4$ have an odd and even number of coins respectively, $a_5$ also has to have an even number of coins. Since $a_4$ and $a_5$ both have an even number of coins, $a_6$ has to have an odd number of coins. Similarly, $a_7$ has to have an even number of coins, and $a_8$ an even number as well. But this contradicts above.