You have to specify validation group for each Section. Try this.;
<html xmlns="http://www.w3.org/1999/xhtml">
<head runat="server">
<title>Untitled Page</title>
</head>
<body>
<form id="form1" runat="server">
<div>
<table>
<tr>
<td>
<asp:Panel ID="Panel1" runat="server" GroupingText="Teacher Login" Width="339px">
<asp:TextBox ID="TextBox1" runat="server" ValidationGroup="teacher"></asp:TextBox><asp:RequiredFieldValidator
ID="RequiredFieldValidator1" ErrorMessage="Please Enter teacher user name" ControlToValidate="TextBox1"
runat="server" ValidationGroup="teacher">*</asp:RequiredFieldValidator>
<asp:Button ID="Button1" runat="server" Text="Teacher Login" ValidationGroup="teacher" />
<asp:ValidationSummary ShowSummary="false" ID="ValidationSummary1" ShowMessageBox="true"
runat="server" ValidationGroup="teacher" />
</asp:Panel>
</td>
<td>
<asp:Panel ID="Panel2" runat="server" GroupingText="Student Login" Width="337px">
<asp:TextBox ID="TextBox2" runat="server" ValidationGroup="stu"></asp:TextBox>
<asp:RequiredFieldValidator ID="RequiredFieldValidator2" runat="server" ControlToValidate="TextBox2"
ErrorMessage="Please Enter Student user name" ValidationGroup="stu">*</asp:RequiredFieldValidator>
<asp:Button ID="Button2" runat="server" Text="Student Login" ValidationGroup="stu" />
<asp:ValidationSummary ID="ValidationSummary2" runat="server" ShowMessageBox="true"
ShowSummary="false" ValidationGroup="stu" />
</asp:Panel>
</td>
</tr>
</table>
</div>
</form>
</body>
</html>