Gray Dogwoods sucker so much that they end up growing in a dense thicket. I think you could plant them 2 feet apart and be OK. In my fields the gray dogwoods stems are about 6 inches apart.

I'd expect Elderberry to grow taller that Gray Dogwood and not sucker or spread as much or as rapidly. I would give the Elderberry some separation from the dogwood - don't plant a dogwood within three or four feet on either side of an Elderberry.

One drawback to Gray Dogwood as a screen is that they are deciduous - in the winter they won't provide much screening. Same goes for Elderberry, which are even less effective as a winter screen because they have far fewer stems. I think you will see lots of birds eating the berries of both shrubs. Both shrubs will fruit much better in full sun than in shade. I have Gray Dogwood growing throughout the woods, but see little fruit on the shade-growing dogwoods. Same goes for Elderberry. Both need about half sun to grow much fruit. If you have lots of shade you should consider a different shrub. Mapleleaf viburnum (Viburnum acerfolium) would be a good choice, although it doesn't grow upright like Gray Dowood - it tends to sprawl.

Thank you, ladyslppr. Deciduous is okay; in the winter my compost area is cleaned up nicely--in the spring and summer I've got stuff piled up all over! Maybe you can picture this for me and imagine if might work: there's already a small red oak planted in the middle of where I want the screen, but it will be years before it casts any significant shade. The lot line runs north-south, there will be sun most of the day except for late afternoon. I was thinking of planting the bushes 4' apart, staggering them a bit, so it would go: dogwood - elder - dogwood - oak tree - dogwood - elder - dogwood. Sound reasonable or too crowded?

Four foot spacing is rather close, but it will help create a screen quickly. You should have a pretty thick row of shrubs in a couple of years. I think your plan is fine.

If I was trying to grow really nice specimen shrubs I would space them farther apart, but for a screen I think you have the right idea.

How far can I plant dogwood trees apart from each other?

What kind of dogwood trees? Different types grow to different sizes.

I like most of Ladyslppr's comments, although if it were me, I'd go at least 6 ft. apart on the grays. So much suckering, no need at all to go any closer. I like gray dogwood despite its somewhat thuggish nature, the fall fruiting panicles in particular being a delight to look at. But remember, this is an ultra-easy shrub, one which will take over a site if you let it. And I think if I was doing this, I might group the elderberries together a bit so as to give them more presence, owing to the paucity of stems, as Ldslpr already mentions.

+om

There's some excellent knowledge in these comments, but the only thing I would offer is that we can't always plan a landscape like it were a room full of furniture. Nature has its own ways. Sometimes the best approach is to introduce variation, redundancy, and randomness in equal measure. We see so many sad stories of gardeners on these boards fighting to save specimen plantings that Nature didn't want in those spots for whatever reason.