From how you mention the Allspice, I assume you would rather have a loose-leafed variety of plant. Not something that will give privacy.
The allpsice will flower and for that reason will have to be given space at the fence---otherwise the continued plucking of flowers off the plant will surely result.
Why not think evergreen--so it doesn't bloom, it gives a lot more privacy and, since you can get to the other side of the fence, can see to its being kept clipped.

Yew--makes a fine, dark green hedge whose needles will never be considered harsh when people walk into it.
Except for spring growth, it is not a hedge that will need constant clipping--it grows more slowly but once established can give a nice hedge that can give privacy next to a fence that I assume is about 4' - 5' high.

If not evergreen, how about a deciduous type---mock orange, privet or alpine currant.
You didn't identify the location you live in so it is difficult to suggest something hardy to any particular area.

kerria is shade tolerant. also some of the scrubby dogwoods might suit the site. need a bit more info about your site. length and width of desired planting. hardiness zone. do you want privacy? or are you screening the fence only? part of the year or year round.

diggerb

If it's near a lake and a wet area then poplars and willows would be good. They grow really fast and should cover that fence in no time.

Oh my, don't plant poplars, they are invasive, invasive, invasive. They're not considered a native and will send out suckers and seedlings. Perhaps you could check out "Bluestone Perennials" on-line and use their search area; putting in your particular needs.