For self fertile varieties, one flower can pollinate itself, a second flower on the same plant is not necessary. For NON self fertile varieties, a second plant/variety is needed. A clone or a cutting of the same plant will not be a viable pollinator. A second variety used to pollinate a self fertile variety will increase the size of the fruit.

The female and Male parts of a same flower are ready at the same time....so one flower is enough !! But the cultivars must be self-fertile !!

Are they any indications as to which ones will be self fertile and which will not? E.g. red fleshed ones tend to be self fertile while white fleshed ones tend not to be.

As a generality, white fleshed DF are self fertile, some of the pink fleshed are self fertile and the red fleshed are not self fertile. There are numerous exceptions to this but it is a start.