Such birds do us good, though we no longer take omens from their flight on this side and that; and even the most superstitious villagers no longer take off their hats to the magpie and wish it good-morning.

参考译文：这些鸟对我们是有益的，虽然我们不再从它们的飞翔姿态来占卜吉凶，连最迷信的村民也不再对喜鹊脱帽行礼，祝它早安了。

问题1：take omens from是什么意思？

解答：take omens from  从...得到/获得预兆/得到前兆

问题2：on this side and that是什么意思？

解答：on this side and that side 意思“在这边和那边”即鸟的飞行姿势.

备注：their flight on this side and that side.  on this side and that side为介词短语作后置定语修饰flight，their为前置定语修饰flight. their flight on this side and that side字面意思为“它们在这边和那边的飞行” 意译为“它们的飞行姿势”。

以上句子整体语法结构分析如下：供网友参考

Such birds【主语】// do【谓语动词】// us【间接宾语】// good【直接宾语】//, though we no longer take omens from their flight on this side and that; and even the most superstitious villagers no longer take off their hats to the magpie and wish it good-morning【让步状语】//.

【主句】:Such birds do us good 这些鸟对我们是有益的，

【从句】:though we no longer take omens from their flight on this side and that; and even the most superstitious villagers no longer take off their hats to the magpie and wish it good-morning.

虽然我们不再从它们的飞翔姿态来占卜吉凶，连最迷信的村民也不再对喜鹊脱帽行礼，祝它早安了。

以上让步状语从句语法分析如下：供网友参考

though【连词】//  we【主语(1)】//  no longer【状语(1)】//  take【谓语动词(1)】//  omens 【宾语(1)】// from their flight on this side and that【状语】// ; and 【连词】// even the most superstitious villagers【主语(2)】//  no longer【状语(2)】//  take off 【谓语动词(2)】// their hats【宾语(2)】//  to the magpie 【对象状语】// and【连词】//  wish【并列谓语动词】//  it【间接宾语】//  good-morning【直接宾语】// .