The personification in this lines is, obviously, that of the tree and the birds. A tree, as an inanimate object can of course know nothing about the birds that have sat on it and it can not miss them when they are gone. In these lines, though, the tree is given the ability to do so.
The effect here is simply to intensify the poems mood of loss and nostalgia. It implies that the speaker, like a tree, has simply stood still, moving from summer to winter. Meanwhile, those people, events, and feelings that made her sing in the summer have gone off like birds leaving a tree.
So the use of this personification intensifies the poem's feeling of the loss of days and people and feelings that have gone and will never again be felt or known in the same way.
The imagery of loss and a barren present seem to permeate the poem. I would suggest the decaying tree help to bring those pictures in full view. The overriding theme of the poem concerns the loneliness of the speaker in the present, contrasting with the shared love experienced in the past. This can be represented quite well with the winter tree which is barren of leaves or fruit, but at one point in time was resplendent in its plentiful quantity of growth. The "winter" that helps to bring this to light is a season of cold, vastness, and a frigidity that is isolating. At one point in time, this tree was full of growth and birds that sing, but now is empty with only its silent boughs present. This personification helps to link with the conclusion of the poem that highlights a summer than once was. In this light, the tree and the speaker are one in the same.
The personification of the tree in the poem 'What lips my lips have kissed' by Edna Millay has the effect of bringing the indoors outside into the natural world. First we were, with the poet, inside the lonely room where we can empathise with her empty unproductive feelings. We then get to travel through the window glass she talks of - she shows us, through the tree outside, the universality of her isolation by explaining to us that there is no relief for her from either inside her own personal space, or anywhere in the countryside or planet - she takes her loneliness with her wherever she goes, like a snail with its shell. She almost 'becomes' the tree and likens its perceived sorrows at losing perching birds to her own loss of loves and companions.