我认为这两种表达都是正确的。《牛津高阶》里均查到了用例：

He was famous, both at home and abroad (= in his own country and in other countries).

The product is sold both at home and overseas.

abroad的英英释义是：in or to a foreign country，overseas的英英释义是：to or in a foreign country that is across the sea。这里的对比很明显了，必须要有sea。一般来说，overseas突出的是距离较远，或是文化、政治等差距大，给人一种较宏大的感觉，abroad仅表示在国外，比较平淡。在这里就看你要不要突出“外国”这个概念了，如果要就用overseas，不要就用abroad。

(1)abroad

adv.  到国外，在海外; 广为流传地;

adj.  往国外的;

n.  海外，异国;

(2)overseas

adv.  在[向]海外，在[向]国外;

adj.  （在）海外的，

从单词构词区别：

(1)abroad=ab+road 意思在道路/大路对面，脱离大路，那就很远，到另外一个国家.  abroad 与道路的陆地有关系，强调在不同陆地，即另外一个国家(land也有国家意思）(ab为否定前缀，意思，非，否，不，脱离）

(2)overseas=over +sea-s，over介词越过/跨过的意思，越过/跨过大海，就是在大海另外一边。与大海有关系，强调在大海另外一端。即北大海分隔的2个不同国家。

以上2种表达都OK,只是at home and abroad  要比 at home and overseas 更常见些

 Mo Yan’s novel Red Sorghum first came to wide attention on the big screen both at home and abroad in 1987.  

莫言的小说《红高粱》于1987被拍摄为同名电影作品并在国内外获得了广泛关注。

Want to talk to your friends both at home and abroad?  

想和你在国内外的朋友交谈 吗 ?

We must grasp the situation at home and abroad.  

我们必须了解国内外形势.

In recent years MNCs have come under heavy criticism at home and abroad.  

近年来,跨国公司在国内外受到猛烈的抨击.

Bids for the new bridge were invited from firms both at home and abroad.  

国内外公司都被邀请对承建该座新桥投标.

Our product command a good market both at home and abroad.  

我们的产品在国内市场上都很畅销.

Situations of preparing isocyanate trimers at home and abroad are mainly introduced.  

主要介绍国内外异氰酸酯三聚体的制备状况.

At home and abroad with high visibility and good corporate reputation.  

在国内外具有很高的知名度和良好的企业信誉.

There are many stories of transfiguration at home and abroad.  

摘要东西方许多民族,都有关于变形题材的故事.

at home and abroad 在国内外(更常用）

She gives frequent performances of her work, both at home and abroad.  

她经常在国内外演出自己的作品。

Agency France Presse enjoys great popularity both at home and abroad.  

法新社在国内外都很有名气.

That country is beset with difficulties at home and abroad.  

那个国家目前内外交困.

Saga features trips for older people at home and abroad all through the year.  

“传奇”旅行社为老年人提供全年的国内外旅游服务。

The firm has identified 60 potential customers at home and abroad.  

该公司已在国内外找到60个潜在客户。

(1)abroad 到国外；在国外 (强调生活在非本国国籍的国家，通常有大海或者海洋把2个国家分隔开来）

I would love to go abroad this year, perhaps to the South of France.  

我今年很想出国，可能是去法国南部。

public opposition here and abroad. 

国内外公众的反对

(2)overseas（在）海外的；（在）国外的 (强调与海外国家有关的事物或者来自于非本国籍的人，通常有大海相隔）

He has returned to South Africa from his long overseas trip.  

他结束了漫长的海外旅行回到南非。

overseas trade figures.  

海外贸易额

Every year nine million overseas visitors come to London.  

每年有900万外国游客来伦敦观光