As Muslims, we know that gambling is forbidden, and we also know that any instruments that have been made exclusively for gambling is also haram, whether betting is involved or not.
As for chess, there are numerous narrations from Ahlul Bayt (a.s.) that speak of chess in a negative reprehensible way. These traditions refer to chess as being bad not in itself as chess, but for 'gambling'.
This is where each Jurist, according to their Ijtihad, derives whether the ruling is applicable to chess and its prohibition, exclusively, regardless of betting on it or no. Or, if it is free of any betting and not seen as gambling at all, then it would be permissible.
Some say that chess is viewed as a mind game, and not as an instrument of gambling. Jurists who see chess in this particular way allow chess to be played, whereas there are jurists who, because of the many narrations, view chess itself to be haram.
The important point here is that this is similar to any other matter in fiqh, where each individual must refer back to their Marja' taqleed for their details and fatwas.
And Allah knows best.